the formulation and analysis of the nine-point finite

Retrospective Theses and Dissertations Iowa State University Capstones, Theses andDissertations

The formulation and analysis of the nine-pointfinite difference approximation for the neutrondiffusion equation in cylindrical geometryHamza Khudhair Al-DujailiIowa State University

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XeroM University Miemfiims 300 North Zeeb Road Ann Arbor, Michigan 48106

76-1816

AL-DUOAILI, Hamza Khudhair, 1939-THE FORMULATION AND ANALYSIS OF THE NINE-POINT FINITE DIFFERENCE APPROXIMATION FOR THE NEUTRON DIFFUSION EQUATION IN CYLINDRICAL GEOMETRY.

Iowa State University» Ph.D., 1975 Engineering, nuclear

X@rOX UniVBfSlty Microfilms, Ann Arbor, Michigan 48106

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEiVED.

The formulation and analysis of the nine-point finite

difference approximation for the neutron diffusion

equation in cylindrical geometry

Hamza Khudhair Al-Dujaili

A Dissertation Submitted to the

Graduate Faculty in Partial Fulfillment of

The Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Department; Chemical Engineering and Nuclear Engineering

Major; Nuclear Engineering

Approved;

In Civ&rge of Major Work

For the Major Department

Iowa State University Ames, Iowa

Signature was redacted for privacy.

TABLE OF CONTENTS

NOMENCLATURE iii

I. INTRODUCTION AND LITERATURE REVIEW 1

II. THEORY 4

A. Neutron Diffusion Equation 4

B. Formulation of Finite Difference Equations 5

C. Application of Green's Theorem on the Leakage Term 8

D. Evaluation of the Line Integrals 9

E. Evaluation of the Surface Integral 24

F. The Five-Point Formula 35

G. The Analytical Solution 36

III. RESULTS AND DISCUSSION 42

A. Completely-Reflected Cylindrical Core 125

IV. CONCLUSIONS 148

V. SUGGESTION FOR FURTHER STUDY 149

VI. BIBLIOGRAPHY 150

VII. ACKNOWLEDGMENTS 152

VIII. APPENDIX A: TAYLOR SERIES EXPANSIONS OF THE

FLUX ABOUT A POINT (r\,Zj) 153

IX. APPENDIX B: SIMPLIFICATION OF DERIVATIVE TERMS 158

X, APPENDIX C; APPLICATION OF THE IDENTITY

? V ^ = _ (j) 174

NOMENCLATURE

b = reflector thickness

c = symbol for the core

e = symbol for the reflector

i = subscript for radial coordinate (i = 1,2,3,4,

j = subscript for axial coordinate mesh points (j

2,3,4 )

r = radial coordinate

z = axial coordinate

B = h = z . - z . , , l o w e r a x i a l i n c r e m e n t : 3-^

D(r,z) = diffusion coefficient

H = core height

L = h = r. - r. ,, left radial increment ^i-1 ^

M = rf - (r^ + R/2)^

P = r? - (r^ + R/2)^

Q = r? - (r^ + r/2)^

R = h^ = - r^, right radial increment

T = h =z. . , - 2 . , top axial increment Zj ]+l ]

B^ = material buckling

H H = # + b

I = modified Bessel's function of the first kind of o

zero order

J = Bessel's function of zero order o

K = modified Bessel's function of the second kind of G

zero order

= modified Bessel's function of the first kind of

first order

= modified Bessel's function of the second kind of

first order

Lg = diffusion length in the reflector

M' = (r\ - L/2)^ - rf

P' = (r^ - 1/2)2 _ J.2

Q' = (r. - L/2)^ - r^

R' = core radius

a = (2.405/R')l/2

V = average number of neutrons released per fission'

Z(r,z) = vEj

(}>(r,z) = neutron flux

2^(r,z) = macroscopic absorption cross section

= macroscopic fission cross section

= macroscopic absorption cross section in the core a

= macroscopic absorption cross section in the

reflector

= neutron flux in the core

(j)g{r,z) = neutron flux in the reflector

I. INTRODUCTION AND LITERATURE REVIEW

In most practical cases, it is difficult to find an

analytical solution to the mathematical problems encountered

with reactor physics calculations. In such cases, it is

necessary to satisfy appropriate boundary conditions on the

boundary of a specified region and to match the continuity

requirements within that region. Analytical methods work in

certain special cases but are of no practical use in many

cases. However, many devised numerical methods have been

developed to deal with such cases, especially for the complex

reactor configurations where there is almost a continuously

varying distribution of materials in the reactor.

In one group theory, the static neutron diffusion equation

in two spatial dimensions for a critical reactor may be

written [1] as

V*D(r,z)V(t)(r,z) - Z(r,z)^(r,z) = 0

w h e r e l { r , z ) = E ( r , 2 ) - v l f i r , z ) a. T

The above equation may be solved approximately by finite-

difference methods which could be obtained using a Taylor's

expansion method or a variational method [2].

In this numerical approach, the technique is to replace

the continuum model by a discrete model, i.e., to replace the

linear partial differential equation by an algebraic system of

simultaneous linear difference equations. In a typical

practical problem these may require a set of 2000, or possibly

more, simultaneous linear equations which must be solved using

a large digital computer [3].

The five-point [2] and the nine-point [4,5,6,7] finite

difference equations for the static neutron diffusion equation

have been formulated in xy-geometry. Also, the five-point

finite difference equations for the time independent neutron

diffusion equation have been formulated in r-z cylindrical

geometry for equal and nonsqual spacing [3,-1,8] »

In the present work, the nine-point finite difference

equations for the static neutron diffusion equation, con

sidering unequal spacing in r-z cylindrical geometry, have

been devised. The nine-point grid of Fig. 1 has been chosen

[ 6 ] in order to expand the neutron flux at the corners about

the point (r^.z^). Another nine-point grid is possible but it

requires one to impose additional continuity conditions to

meet the cases where some of the derivatives are not continuous

across interfaces. However, important aspects to consider in

the formulation of the nine-point technique is to insure

higher accuracy for the solution of the finite difference

equations.

The strategy of solving the finite difference equations

depends specifically on the iterative scheme. For elliptic

systems, the iterative schemes may be categorized as stationary

and nonstationary iterations. However, the main predominant

methods in solving elliptic difference equations are successive

overrelaxation (SOR) iterative techniques and alternating-

direction-implicit (ADD iterative techniques [9]. In this

work, successive overrelaxation has been used without any

attempt to optimize the SOR parameter.

II. THEORY

A. Neutron Diffusion Equation

The basic steady state neutron diffusion equation in a

nonhomogeneous medium is:

"V»D(r, z) V(() (r, z) + Z (r, z) (|) (r, z) - S(r,z) = 0 (1)

The leakage term can be expressed in cylindrical geometry,

r-z coordinates, as

•V*DV(j) = - — 1 r 3r D* r 3i 9r 9z

Insertion of relation (2) into Eq. 1, results in

1 9 r 3r

- M 9z

+ E (j) — S = 0

In the absence of extraneous source, the equation after

multiplying both sides by r may be written as

9_ 3r D*r

9 ({) 9r 9z

- rE(J) = 0 (3)

In this case any fissioning occurring in the medium has to be

contained within the term of Eq. 3. Thus, Eq. 3 is a

homogeneous equation.

B. Formulation of Finite Difference Equations

The one-group neutron diffusion equation in two spatial

variables is considered over region R in a physical system

such as a nuclear reactor. This region contains a homogeneous

finite-connected number of sub-regions, R^, (n = 1,2,3,4), as

shown in Fig. 1. The increments and the coordinates of the

points in Fig. 1 may be written as follows:

1. Increments 0(h):

fi+l - =^1 ' = R Right

L Left i-1

Sj+l - Zj = hz. = T

B Bottom

2. Coordinates

o = [r^^ , Zj]

c = [r\ , Zj + |]

^ ~ ^^i "• 2 '

Fig. 1. Mesh structure in the interior

The region parameters D^, and are material property

parameters which are nonnegative and piece-wise continuous in

sub-region, R^. Discontinuities do exist at interfaces between

dissimilar media. However, the parameters are usually

constants within the region R and the neutron flux and its

derivatives are continuous over the region, R, [2]. Thus,

these assumptions allow one to perform the surface integral of

Eq. 3 for the region R, over the volume of a reactor generated

by rotating the rectangular region R about the z-axis (2wrdrdz)

- rZ^O(r,z) drdz = 0 (4)

Consequently, the previous assumptions permit the application

of Green's theorem in order to transfer the surface integral

into a line integral around the boundary of the sub-region

oriented in the direction as shown in Fig. 1. Hence, Eq. 4

may be written as

ar 3r + 3z

M 9z drdz -

r$(r,z)drdz = 0

C. Application of Green's Theorem on the Leakage Term

Green's theorem can be applied in order to convert the

leakage term of Eq. 5 into a line integral. Green's theorem

may be expressed as [10]

drdz = [M{r,z)dr + N(r,z)dz]

N(r,z) = r 30(r,z) 9r

M«r,-z) -r-9z

Thus, the leakage term of Eq. 5 may be written as

I D n=l n

^ _ m 3r 3z drdz = Id

n=l n [Mdr + Ndz] (7)

K, n n The right hand side of Eq. 7 may be expanded as

n=l n [Mdr + Ndz] = D, [--r dr + r §& dz

+ D, -r 14- dr + r 14- d^ + D d z 3r -r §$ dr + r dz

+ D, - # + r M dz" ( 8 )

D. Evaluation of the Line Integrals

The line integrals in Eq. 8 can be evaluated using the

following techniques:

1. The neutron flux, #(r,z), has continuous partial

derivatives in a neighborhood for the point

^^i'^i^ in the (r,z) plane. Thus the neutron flux

can be expanded about the point using a

Taylor's expansion for functions of two variables [81

3 4(r,z) = Z ^ [(r-r.)D + (z-z.)D ^){r.,z.)

m=o m! ^ ^ ^ ^ ]

v ( r , z ) = ' - " - j - ° z * o

+ (r-r.) (z-z.)D^2V

+ ^^r-r^)2(z-Zj)Drrz4o + j) + O(h^)

(p, . = (})(r. ,Z.) = , D = , D - ^ i,j - - Vo# " 9r ' z " Sz ' rz " 9r9z '

^rrz " 3r3r9z ' ^rzz ~ 3r9z9z

2. The partial derivatives, and

can be obtained by differentiating the flux (plr,z)

in Eq. 9 with respect to r and z, respectively, that

+ lr-fi)D2*o + (:-Zj)Drz+o

+ i (r-r.)(z-ZjlDrrz*o

+ I (Z-Zj)\zz^0 + O(h^) (10)

+ J (r-ri)^Drr2*o +

+ I (Z-Zj)2 + O(h^) (11)

Therefore, and can be evaluated at the

points (r^ + z-zy), (r^^ - z-z^), (r-r\, z + j), and

g (r-r^, Zj - gj. The results are

9(|) 3r

= + I + (z-SjlDrzt,

+ f Dr*o + I '^-^j'°rrz*o

+ J lz-2j)2 0^22*0 (12)

2 = Dr*o " I Dr*o (2-Zj)Drz*o + I"

^ ^i'2

- I (z-ZjjDrrA + 5 Drzz^o

3* 3z = D,*o + lr-ri)Dr:*o + | + 5 U-r.)^

Z=Z.+j

+ I (r-r. ) Dp22*o + |- Dz*o (")

a 3z = + (r-r.jD^^'f'o " I + I (r-r^)^ Dfrz^o

z=z,-f

- I (r-r. ) 0^22*0 + f- "z't'o 'IS)

3. The continuity of the neutron current density can be

applied at material interfaces in order to eliminate

the fon-jard and backward line integrals along the

same paths. This may be visualized as follows:

d c j ) 9z dr - D,

= - D, 3^ 9z

dr + D, R,

ii 9z dr

But D, 9 (}) 3z

= D. Ri

92 because the neutron current density is R,

a continuous function across material interfaces. Thus,

r[ dr + D, R,

dr] = 0. Similar relations R,

occur along (o,c), (o,f), and o,k).

Therefore, the line integrals in Eq, 8 may be evaluated

by performing the integration along the paths L^, Lg, and

as follows:

3* 9z T

:==i+2 dr

In Appendix B are developed the expressions for these various

partial derivatives. These expressions are then further

algebraically simplified. By substituting for 9(j) 9 z T and

3 (}) 9r

from Eq. 11 and Eq. 4 of Appendix B, one has

D, B4W dr + r 3 (j) 9r dz

= -D, r[G + (r-r^)H + (r-r^) K]dr

+D, r[A + (z-z.)F + (z-zj)^c]dz

-2DiG[ri-(ri+2) ]^ [r^-{r^+2) ~2^i ^

+DiA(r.+|)-| + D^F{r.+|)-|- + Di(v|)C- — (16)

Define:

P = (ri + §)2

Q = r. - (ri + |)

M = 4 ^i (ri + f)

By substituting for G, H, K, h , F , and C in Eq. 16 from

Eqs. 8, 9, 1, 2', and 3 of Appendix B and collecting all the

4 terms of orders 0(h ) and higher under the truncation error

0(h ), one has

O" .(D

eu Q J N -e-N II N

tolt-3

i—n M

O) QJ N -e-

Cb lJIJ

lo 1--' •«=•1 o

LiJ D t-{

N N •e-

ïd to

h oj -©•

WI-' œl O M

t-3 to

•D O

N 00 -e-

o to m-

H-" -4

sa oo| f-lO I

to|t-t-s

+ tOlM

ÎO O —i 1-3

ii-ait 00|M lO I

tOlM l-j

D H3 M 1-0

QJ H0-

Oi l-i + M

os •e-

Oi N I

I col to

t-i H-lO +

ton-* K

H- to ^0

œ| 1-3

-t-tolïO

o M H N

By substituting for 3^ 9z

and I % fa* ij 13r

'j 2 Z=Zj +

Eq. 11 and Eq. 7 of Appendix B, the result is

L r=ri-2

• [9z

dr + r M4

^i-l f

-Dr r[G + (r-r^)H + (r-r^) K]dr

r^i +D, r[A' + (z-Zj)F' + (Z-Zj) C]dz (18)

Define:

P » = ijv 4 * 2' 4

Q' = (r, - L. 3 -) - r.

M' = '^1 4

By substituting for G, H, K, A', F', and C in Eq. 18 from

Eqs. 8, 9', 10, 5, 6, and 3 of Appendix A and collecting all

the terms of orders O(h^) and higher under the truncation

error 0(h ), one has

D, i wm

dr + r llldz

1 DgP'

2 — (*2 -

i Q- - i r.P- + Î • I ( ( j ) - ^ 3 - # 2 +

^ 1 °2''

* 2 ~ (4>3 - tgl

5 "2 I ^ i O ' + 1 4 ^ '

+ L I Q' i fi?' ^rrz'l'o

+ k »2?'ri°r::*o

D, -r ISl"

= -D. 3 (j> 9z z=z^ B

+ D, dz

-"-i"2

By substituting for ^ and

Eq. 7 of Appendix B, one has

3<j) 9r r=r

^ from Eq. 4 and

-r 9(j> 32

dr + r 9 (j) 9r dz

r[G' + (r-r\)H' + (r-r^ ) Kjdr

+D, r[A* + (z-z.)F' + (z-z.)^Cjdz ] J

( 20 )

By substituting for G', H', K, A', F', and C in Eq. 20 from

the Eqs. 12, 13, 10, 5, 6', and 3 of Appendix B and collecting

all the terms of orders O(h^) and higher under the truncation

4 error 0(h ) one has

D, -r 9z dr + r||^ dz!

1 V Z is

((t.» -•o'

i Q' - i r.p'

+ & D, M. - I r.Q- + i r^p.j

+ L I 0' - I V Drrz*o

+ 5? (21)

[-l»l dr + r 3r dz

= - D, ii B dr + D4 :==j-2

r=ri+f dz

By substituting for _ and [|^] „ from Eq. 14 l3:;z=z.-2 l::^r=ri+#

and Eq. 4 of Appendix B one has

[j:: (%)

r[G'+(r-r^)H'+(r-r^) Kjdr

+ D, 'i

r[A + (z-z.)F + (z-z.) C]dz J J

( 2 2 )

By substituting for G', H', K, A, F, and C in Eq. 22 from the

Eqs. 12, 13', 10, 1, 2, and 3 of Appendix B and collecting all

the terms of orders O(h^) and higher under the truncation

error O(h^) one has

D. [3z

dr + r 9(f) 9r

1 *4^ A ) - 2 -B- (^4 - *o)

R ïNi 0 - & fi? B [r + Ê 8 i 2

((j)^ - (j)^ - (j)^ + (f)^)

(<l>i - •o>

- Is DaBsfrpDfOo

+ 1 °4 Î M - I r.fl + I r^P -R §0 & fl? ^rrz'f'o

i »4:iB'»rzz+o ( 2 3 )

By substituting for the terms that contain the derivatives in

the Eqs. 17, 19, 21, and 23 from the relations 25, 33, 36, and

39 of Appendix B one has

E D. n=l n

+1- fr #1 3z drdz

1 DiP 2 — (^2 - *o)

R & Q - & fi» )-| ri + I (4) - - Oo + (|)n)

1 V + 2 — + f (4)1 - •o'

& D^r.RT(TD3^„ + RD&.)

24 Drrzto ^ '^rzz'^o^

1 V - 2 " T — " *o)

I f l Q ' - I V I k - ((j) - - On +

^ i ¥ k - i (*3 "

Is Ojr.LTtTD^ - LD3*o «2

h Dar. (L^ - T3 D„,4.„>

1 DgP'

2 B (4) A -

D. I f l B 3 Q i r . P - + I (<t> " 4^ - ^4 +

(*3 - *o)

+ 38 »3 fi + LDr*o) R.

+ L Djr. (L^ D,r:*o + R,

1 D4P - 2 -B- (*4 - *0)

R i Q - & fi? B 8 + I (t}) - <t>i - (f>A +

(4)1 - *0)

+ 1 8 '®°z o - o' R,

* h Vi'^Xrz'fo -(24)

2 In Appendix C, the identity V ~ and a Taylor's expan-

sion are applied to perform further simplification on the

derivative terms of Eq. 24. By substituting for these terms

in Eq. 24 from Eq. 31 of Appendix C, one has

Z D_ n=l n

iil + A_ fr Iil 3r, 3z ^ 9z

1 V 2 T

D, 1 T

{ ^ 2 -

i 0 - T fi? T fr + 8 .^i 2. (4) " 4^ - $2 *

44- h (<i>, - •o>

1 D.P' i -f- (+2 -

1 0' - è fi?' + I fr - k] + 8 i 2. ((j) - 4^ - ^2 + <)>q)

+ 1 _^2l Ir, _ t 2 L ri 2

1 D,P'

t T™ '^4 " *o'

'3 • *o)

i Q' - i r.P'] + I [r. - I] (()) - - #4 + 4»^))

+ 1 *3» + 2 — " 2 (*3 " *o)

1 V 2 B

I r i •" t ((j>^ - - *4 + (})q)

, 1 ''4® * 2 — ri + Il (4.^ - •„)

^ Î8 °1 °1 <•! + *2 - 2*c)

" 48 "2 "2 "^i '•''' (*2 *3

- èê °3 "3 (*4 + *3 - 2*0)

48 °4 "4 '*4 + *1 - 2*0'

1 T + JR (R - L) [LD^tOi - (t)^) + RD2((i>3 - *0)]

1 B + *4 LR (* - L) [LD4(*1 - *0) + R°3(*3 " ^

+ 36 ER (L + R)[LDi(4^^i^-*2+*o)-BD2(*2-*3-*2+*Q)]

+ Ir (L + R)[LD4(**-*i-44+*o)-*D3(*^-*3-*4+*o)]

1 r-R 1 A + 2% TB~ [0^8(4 -4'2-42+0o)^^4^(^

ir Xj ^ I4 TI~ l:D2B(4f-*2-*2+*o)+D3T(*3-*3-*4+*Q)]

+ §4 L&- [DiL(4^^*i-42+*o)+D2R(*^-*3-42+*o)]

+ O(h^) (25)

E. Evaluation of the Surface Integral

Consider the surface integral term

ij n=l J J r 0(r,z)drdz of Eq. 5.

In this case, the Taylor series for the flux expansion is

truncated to 0(h ) since the truncation error will be of order

O(h^) after performing the integration. The flux expansion is

(j)(r,z) = + (r-r^)D^*Q + (z-ZjiD^^Q

Thus, by inserting this flux expansion into the above integral

term, one has

4 - I n=l

r I(j)^ + (r-r\)D^*Q + (z-Zj)D^())^ldrdz

The integration may be performed as follows:

a: For n = 1 rp T3

- E, r[*o + (r-r^iD^^Q + (Z-Z .)D^(j)j^ ] drdz R,

î + I El? i 0 - & fi? »r*o R,

( 2 6 ) R,

For n = 2

r[*o + (r-:i)Dr*o + (z-ZjiDg^o ] drdz R,

L z_. i "2 r.-TT -]

= T Z; ? P' *o + & Z; %(& 0' -I V, °r*o

+ 16 ^2 F' »z»o (27) R,

For n = 3

z. r. ] 1

rl*o + lr-ri)Dr*o + (z-z.)D 6 Idrdz J z o _

y 2 "i 2

= î 3

+ i £,Bf4 Q' - t rjP'lD,^ J 6. u. j ; ' °'R,

16 E,»' P' °z»o ( 2 8 )

d: For n = 4

z. r. +?

r ] 12

r[*o + (r-r^)D^(j)^

B ^j"2

R, + (z-zy)D,*o ]drdz

i £4 B P + i Ï4 B i Q - I ri P »r+o

& Z4 s' P »:*o (29) R,

The Eqs. 26, 27, 28, and 29 may be simplified by substituting

for P , (3 Q - I r^P , P', and i n» - i 2 r\P' I from Eqs. 15,

18, 26, and 29 of Appendix B. All the terms of orders 0(h )

and higher are lumped under the truncation error O(h^). Hence,

Eqs. 26, 27f 28, and 29 may be written as

- 1. r(fj{r,z)drds = = T P

- k E. T r.R^ R,

T6 (30)

r (()(r,z)drdz I Zj T p-

+ & 2 °r*o

Î6 ^2 Oz^o R, (31)

'j"2 ^i"2

r (t)(r,z)drdz = + ? %3 B P' •o

+ L Z3 Br.

+ 16 L Dj+o R.

r ({)(r,z)drdz = + Z^ B P (|)^

^R ^j'2 ^1+2

^ E, r. BR:

+ lê fi Dz*o R,

By adding Eqs. 30, 31, 32, and 33, one obtains

r*(r,z)drdz = ^ T P 4 ft

- I n=l

Î6 Or*o R,

k Î1 RT^

+ T Ez ?! * o

+ 16 C; fi »r*o Rr

Î6 ^2 =^1 °z*o Rr

+ t %3 B P' *o

If =3 Bfi Orto R.

+ h h ^ °.*o R-

+ T Z4 B P *0

h h =^1^ °r*o R,

+ riR D^to R,

Substituting for the terms that contain the derivatives in

Eq. 34 from Eq. 41 of Appendix C, one has

4 - E n=l r(() {r,z)drdz = ^ T P

+ i 2 T P' *0

+ i Sj B P' *0

+ T 4 B P *o

- Î 6 + * 2

I6 ^2 + *2 - 2fo>

- b -3 "i'-'-h + i'À - 2^,1

- & Î4 BfiRt*! + n • 2*o'

+ O(h^) (35)

Finally Eq. 5 may be expressed in a finite difference form

by adding Eq. 25 and Eq. 35. The result is

.!A 3_ 3r li(^ Ifl

4 drdz- E

n=l, E^r(l)(rjz)drdz = 0

I WlH* ' '—:—» w|l-' O I

ton-* M

H-•O

oo|tz)

toll-' O •I& tx>

tojM I O tr" uj

fi a W

toll-'

tojtr'

WlM W| H-" lO

N)| hi

to O W ^0

•e-iP>

tOlM 1 o tr* to I t-3

toltr*

o col CO

•(> (a>

-<?• W

-O-il&

tr"! O to

tOlM M

oo|t-3

h3 a to

-©• to

•e-o

h3|l WlM o I

tOlM M

oo|i-3

{U 3 O.

tojtr»

-e-to I

-©-w -o-

-e-to -e-

o -e-O

«•l - Ool

• 3^ "^1 "l + *2 - 2*o)

• 58 "2 "2 T<*2 + *3 - 2*0)

' ÏI °3 "3 *(*4 * *3 2*0)

is "4 "4 V B(*4 + h - :*o)

+53- #R(R - - •„) + RD2(»3 - *ol]

+§4 EgtR - L)[LD,(4i - i.^) + RDjC+j - t,)]

+ |ç (R+L) ILDi(*^ - *1 - *2 + <l'o''®®2'*^ - *3 - *2 + •o"

+ 5J |r(R+L) - *1 - *4 + 4iQ)-RD3(*3 - *3 - O4 + *0)]

1 ri*... ,,1 .4 + |j 5i|-tBD^(+ - *1 - *2 + tol+TD,!* - *1 - *4 + *0)1

r.L + 54 ïg-tB°2'4'^ - * 3 - +2 + - *3 - *4 + •o"

Lfil 24 LR

[LD^((j)^ - (|)^ - ({>2 + <t)^)+RD2 ((j)^ - <!>3 - <î'2 4)q) Î

1 ,.3 + 2 4 LR~^^°3^<f' - *3 - *4 + <|)q)+LD4('I) - *1 - *4 + <f>o)l

+ i Î2 T P'

+ I Sj B P'

+ I Ï4 B P +0

& =1 T*(*l + *2 -2+o)

16 ^2 ?LI*3 + *2 -2*o)

- Zj r. BL(*3 + - 2y

- I6 C, ?! **(*1 + *4 -2*o)

+ O(h^) = 0 (36)

The finite difference Eg. 36 may be written in the form

^i,j*o + + =1,3*2 + d.

+ h. .((1^ + kj + O(h^) = 0 1 f ] 1 f J

+ °(h*) = ° (37)

The flux coefficients may be written for the equation

E (r,z) - v Z j{r, 2 ) -V 0(r,z) + Z(r,z)#(r,z) = 0, where Z(r,z) = — D^(r,z)

for a reactor and Z(r,z) = ^ for a reactor reflector,

as follows:

Ki,i = -°i,i-li 2hJ (3r. + h^ ) + 3hJ (2r. + h ) î ^ î =i-l ^ î

1 (h 'j-1 "i

, ] n vii V n ) + 4r. (h^ + ) l/96h h z- •. r, 1 r^ Zj-i f ^i ^j-i

hi,i = ^ 2hJ (3r. - h^ ) + 3h; (2r. - h ) Vl ^ ^i-1 =i-l ^ ^i-1

- hi i\ + iV ) + 4r. (hj + hi ) >/96h h ""i î-l ^ î-l Sj-l f î-1 Zj-l

2h h ) + 3h' (2r. - \.l'

h , j ' : 2h^ (3r. + ) + 3hl (2r. + h ) Ti 1 r. z j 1 ^

+ + h > + 4^1 (hfi + "l' j 1 1-1 1 J

®l,j = -hi,j - ki,i -h_ )

+ '°i,j-l''r. - Di-l.jhri.i' /8h

®i,l = -9l,i - hi.i " ''\.l +

j = - 9i,i

(Di,jhr. - °i-l,j^r._^^ /8h 'i

. = - f. . - k. . - (12r. + 7h - h )(D. .h lO If] ri r._^ 1,3 2-

- <"1,3 °i,j + *1,] + ®i,j + fi,i + hi.i + 9i,i

" _ )(2r^ + h )/16 1 3 j" J- ^

F. The Five-Point Formula

The five-point formula in r-z geometry can be written as

bi,i = ^ "r.'

®i,j = -'°i,j-l\

*i,i ' -<bi.i ^ °l,i + *1,] " ®i,j'

+ {h (4ri + h )(ï._.h + ïi,j-ih. ) X X J J J-

G. The Analytical Solution

The analytical solution of the neutron diffusion equation

over regions with material interfaces is difficult to be

obtained even in the simplest cases [5]. To assure the cor

rections and accuracy of the numerical solution of the finite

difference equations, it is useful to solve the neutron dif

fusion equation analytically over the region R for certain

model problems. This technique gives a suitable comparison

between the numerical and analytical solution of the neutron

diffusion equation. With this comparison, the error range

between the two solutions can be determined.

The distribution of neutrons in the core and reflector of

a one energy group system may be suitably described by the

following equations [11]

V^(j)^(r,z) + B^(r,z) = 0 (39)

V^({)g(r,z) - = 0 (40)

®c D: -

^4 = a

e z Reflecto:

C = Core

The analytical solutions of Eqs. 39 and 40 for radial and

axial reflectors in cylindrical geometry (Figs. 2 and 3) have

been determined [11] and may be written as follows:

a: radial-reflector (Fig. 2)

(|)^(r,z) = Oco^o^^c^) 5 (41)

/ l-t-N

Fig. 2. Radially-reflected cylindrical reactor core

<l>e(r,z)

I [a (R'+b)]K (a r)-K [a (R'+b)]I (a r) *CoJo — — — COSjjZ

1 [a (R'+b)]K (a R')-K [a (R'+b)]I (a R')

where is to be determined from the following equation;

Jl(a^E')

a D e e

Iq [a^ (R' +b) ]K^ (OgR' ) +Kq [a^ (R'+b) ] (a^R' )

[oig (R' +b) ](agR')-KQ[a^ (R'+b) ](a^R' )

ttg and can be calculated from the following relations;

i.H, (44)

Be = «C + : (45)

b; Axial-reflector (Fig. 3)

Og(r,z) = cos (46)

cos ( B H / 2 ) « Kefr.z) = sinh(6^b) sinhte^(5 " (47)

REFLECTOR

[REFLECTS

3. Axially-reflected cylindrical reactor

where 3^ is to be determined from the solution of the following

equation:

D I 3^ tan(3^H/2) = 5^ | 3^ coth[Bg(| - |) ] (48)

a and can be calculated from the following relations:

2.405 R'

P (e) a a

where 's. tt

H = I + b

R' 5 Radius of the cylinder

H = The height of the cylinder

b E reflector thickness

III. RESULTS AND DISCUSSION

The results of this work were obtained by solving Eqs, 37

and 38 numerically and Eqs. 39 and 40 analytically using a

computer program. Within the computer program of the analytical

solutions Eqs. 43 and 48 are solved numerically using Newton's

iterative method [12].

The computational technique is concerned with iterative

methods in solving the nine-point and five-point finite dif

ference equations [13,14]. The iterative method, specifically

the overrelaxation method, is used because it is easily pro

grammed and has a comparatively fast convergence rate.

Various cylindrical reactor core types such as bare cores,

radially reflected cores, and axially reflected cores were

considered. The calculated results for the previous configura

tions cover equal (Figs. 4, 5, and 6) and nonequal (Figs. 1,

8, and 9) spacing along radial and axial axes.

The neutron flux values for equal spacing along the r- and

z- axes for the nine-point, five-point, and analytical solution

are tabulated in Tables 1, 2, and 3. The absolute error^ and

^Absolute error = numerical solution-analytical solution.

Euclidean norm for the radial and axial cases are computed as

listed in Tables 4, 5, 6, 1, 8, and 9. Graphs of absolute

error versus radial and axial distances are plotted as shown

in Figs. 10-32. Plots of residual vector norm^ versus itera

tion are shown in Figs. 33-35. From the plots of Figs. 10-32

one can observe the accuracy of the nine-point formulation over

the five-point formulation.

The over all Euclidean norm for the nine-point and five-

point solutions are 0.0053 and 0.1103 (bare cylindrical core),

0.0109 and 0.1433 (radial-reflected cylindrical core), and

0.0075 and 0.01720 (axial-reflected cylindrical core),

respectively. The comparisons between the Euclidean norms for

'The Euclidean norm is [2]

2 ^2 ~ s lEjl

where E^ is the absolute error at a mesh point,

(i = 1,2,3,4 —).

^The residual vector may be defined as [9]

r(P) = - x'P)

where r = the residual vector X = the solution P = iteration number, (P = 0,1,2,3, ).

Thus, the residual vector norm may be defined as [2]

Z (r.) i ^

the nine-point and five-point results illustrate the accuracy

of the nine-point formulation over the five-point formulation.

The convergence rates^ as calculated from the slopes of

the curves Figs. 33-35 for the nine-point and five-point

formulas are 0.654 and 0.448 (bare cylindrical core), 0.464

and 0.304 (radially reflected cylindrical core), and 0.406 and

0.412 (axially reflected cylindrical core), respectively.

These results show that the nine-point formula converges faster

than the five-point formula for these cases. The saw-tooth

effect in the curves of Figs. 34 and 35 represents the inner

iteration over the eigenvalue (Z(r,2)/D(r,z)) of the boundary

value equation until criticality was achieved.

^The convergence rate may be defined as [9]

V = - x.n A2^

V = the convergence rate

= the largest eigenvalue of the iteration matrix.

<P = 0

4> - 0

Pig. 4. Equal spacing mesh for a cylindrical bare reactor core

Fig. 5. Equal spacing mesh for a radially reflected reactor core

Pig. 5. Equal spacing mesh for axially reflected cylindrical reactor core

• z o

Fig. 7. Unequal spacing mesh for a cylindrical bare reactor core

Fig. 8. Unequal spacing for a radially reflected reactor core

Fig. 9. Unequal spacing mesh for axially reflected cylindrical reactor core

Table 1. The neutron flux distribution in a bare reactor core for equal spacing along the radial and axial axes and for input data D = 1.0 cm, Z = 0.33 cm"l, and R' = H/2 = 5.0 cm ° °

Method r

z 0 1 2 3 4 5

ANS* 0 1.000 0.943 0.782 0.544 0.268 0.000

1 0.951 0.897 0.744 0.517 0.255 0.000

2 0.809 0.763 0.632 0.440 0.217 0.000

3 0.588 0.554 0.460 0.320 0.158 0.000

4 0.309 0.291 0.242 0.168 0.083 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000

NPS^ 0 1.000 0.942 0.783 0.545 0.270 0.000

1 0.951 0.896 0.744 0.519 0.257 0.000

2 0.809 0.763 0.633 0.442 0.219 0.000

3 0.588 0.554 0.460 0.321 0.159 0.000

4 0.309 0.292 0.242 0.168 0.084 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000

FPS^ 0 1.000 0.895 0.731 0.506 0.249 0.000

1 0.955 0.854 0.698 0.483 0.238 0.000

2 0.815 0.730 0.596 0.412 0.203 0.000

3 0.595 0.532 0.435 0.301 0.148 0.000

4 0.314 0.281 0.230 0.159 0.078 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000

= analytical solution.

^NPS - nine-point solution.

^FPS = five-point solution.

Table 2. The flux distribution in radially reflected core for equal spacing and for input data of D = 0.5 cam, Z = 0.1385 cm"l, R = z = 5 citi/ Dp = 1.0 cm,

-1 c = 0.1 cm , and b = 2 cm

Method r z 0 1 2 3 4 5 6 7

ANS 0 1.000 0.956 0.830 0.637 0.405 0.160 0.067 0.000

1 0.9 51 0.909 0.789 0.606 0.385 0.153 0. 064 0.000

2 0.809 0. 773 0.671 0.516 0.327 0.130 0. 054 0. 000

3 0.588 0.562 0.488 0.375 0.238 0.094 0.039 0.000

4 0 .309 0. 295 0.456 0.197 0.125 0.050 0. 021 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0. 000 0.000

NPS 0 1.000 0.955 0.830 0.639 0.408 0.165 0. 068 0. 000

1 0.951 0.908 0.789 0.608 0. 388 0.157 0.065 0.000

2 0. 809 0.773 0.671 0.518 0.330 0.133 0.055 0 . 0 0 0

3 0. 588 0.561 0.488 0.376 0.240 0.097 0.040 0.000

4 0.309 0.295 0.256 0.198 0.126 0.051 0.021 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

FPS 0 1.000 0.915 0.782 0.595 0.376 0.146 0.061 0.000

1 0.951 0.870 0.744 0.566 0.356 0.139 0.058 0. 000

2 0.809 0.740 0.633 0.482 0. 303 0.118 0.049 0.000

3 0.588 0.538 0.460 0.350 0.220 0.086 0.036 0.000

4 0.309 0.283 0.242 0.184 0.116 0.045 0.019 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Table 3. The flux distribution in. axially reflected core as for equal spacing and input data D =0.5 cm Z = 0.1536 cm~l, R = z = 5 cm, D_ =1.0 cm,

C C = 0.1, and b = 2 cm

Method z r 0 1 2 3 4 5 6 7

ANS 0 1.000 0.962 0.852 0.678 0.452 0.192 0.082 0.000

1 0.943 0.907 0.803 0.639 0.426 0.181 0.077 0.000

2 Q . 782 0.752 0.666 0.530 0.353 0.150 0.064 0.000

3 0.543 0. 523 0.463 0.368 0.246 0.104 0.045 0.000

4 0.268 0.258 0.228 0.182 0.121 0.052 0.022 0.000

5 CI.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NPS 0 1.000 0.962 0.852 0.677 0.451 0.188 0.081 0.000

1 0.942 0.406 0.802 0.637 0.424 0.180 0.076 0.000

2 0.781 0.751 0.665 0.528 0. 352 0.149 0.063 0.000

3 0.544 0.523 0.463 0.368 0.245 0.104 0.044 0.000

4 0.269 0.259 0.229 0.182 0.121 0.051 0.022 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

FPS 0 1.000 0.962 0.851 0.674 0.447 0.185 0.080 0.000

1 0.891 0.857 0.758 0.601 0.398 0.165 0.071 0.000

2 0.726 0.698 0.617 0.489 0.324 0.134 0.058 0.000

3 0,500 0.481 0.425 0.337 0.223 0.092 0.040 0.000

4 0.246 0.236 0.209 0.166 0.110 0.045 0.020 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Table 4. The calculated results of Euclidean norms for equal different axial levels

absolute errors and spacing in a bare core at

Five-point z r cm cm X 10

•3 . Over all 2 norm

E^ X 10"3 . Over all 2 norm

0 0 0.0000 0.003 0.0053 0.0000 0,082 0.1103

1 0.8356 48.3252

2 0.5273 50.6976

3 1.8940 37.6814

4 1.9957 18.7239

5 0.0000 0.0000

1 0 0.1614 0.003 3.7467 0.012

1 0.4543 42.6084

2 0.7814 45.2221

3 1.9979 33.9430

4 1.9949 16.8740

5 0.0000 0.0000

2 0 0.4057 0.003 6.3860 0.058

1 0.1193 33.3836

2 0.9005 36.1296

3 1.8649 27.2561

4 1.7732 13.5566

5 0.0000 0.0000

3 0 0.4868 0.002 6.9725 0.097

1 0.1120 22.1675

2 0.8245 24.5436

3 1.4754 18.6226

4 1,3523 9.2679

5 0.0000 0.0000

4 0 0.3376 0.0013 4.8964 0.0196

1 0.1650 10.5528

2 0.6459 12.0030

3 0.8389 9,1678

4 0.7424 4.5655

5 0.0000 0.0000

Table 5. The calculated results of absolute errors and Euclidean norms for equal spacing in a bare core at different radial levels

Nine-point Five-point r z cm cm X 10"3 0 Over all

2 norm X lO'^ 0 Over all 2 norm

0 0 0.0000 0.0007 0.0053 0.0000 0.0113 0.1103

1 0.1614 3.7467

2 0.4057 6.3860

3 0.4868 6.9725

4 0.3376 4.8964

5 0.0000 0.0000

1 0 0.8356 0.00098 48.3252 0.07660

1 0.4543 42.6089

2 0.1193 33.3836

3 0.1120 22.1675

4 0.1650 10.5528

5 0.0000 0.0000

2 0 0.5273 0.00167 50.6976 0.08165

1 0.7814 45.2221

2 0.9005 36.1296

3 0.8245 24.5436

4 0=6459 12.003

5 0.0000 0.0000

3 0 1.8940 0.0037 37.6814 0.0612

1 1.9979 33,9430

2 1.8649 27.2561

3 1.4754 18.6226

4 0.83B9 9.1678

5 0.0000 0.0000

4 0 1.9957 0.0037 18.7239 0.0304

1 1.9949 16.8740

2 1.7792 13.5566

3 1.3523 9.2679

4 0.7424 4.5655

5 0.0000 0.0000

Table 6. The calculated results of absolute errors and Euclidean norms for equal spacing in a radially reflected core at different axial levels

Mine-point Five-point

cm cm E. x IQ-: A, E. x lO":

0 0.0000 0.0063 0.0104 0.0000

1 1.1444 41.3011

2 0.0221 47.6269

3 1.8909 42.1592

4 3.6502 29.8813

5 4.3610 14.4312

6 1.5600 5.8537

7 0.0000 0.0000

0 0.0182 0.006 0.1877

1 1.0802 39.1359

2 0.0237 45.1908

3 1.7912 40.025

4 3.4646 28.3779

5 4.1445 13.7596

6 1,48251 5.5611

7 0.0000 0.0000

0 0.0481 0.005 0.3488

1 0.9033 33.1447

2 0.0160 38.3341

3 1.5213 33.9750

4 2.9433 24.0990

5 3.5234 11.6473

6 1.2600 4.7248

7 0.0000 0.0000

0 0.0589 0.004 0.3735

1 0.6458 23.9867

2 0.0092 27.7811

0.0830 0.1433

(Continued)

Nine-point Five-point

1.1033

2.1357

2.5584

0.9146

0 .0000

0.0393 0=002

0.3373

0.0058

0.5777

1.1208

1.34413

0.48045

0 .0000

24.6372

17.4816

8.4523

3.4291

0 . 0 0 0 0

0.2401 0.025

12.5752

14.5782

12.9335

9.1794

4.4395

1.8010

0 . 0 0 0 0

Table 7. The calculated results for the absolute errors and Euclidean norms for equal spacing in a radially reflected core at different radial levels

Nine-point

E. X 10 1

Over all norm

Five-point

E, X 10 1

-3 Over all norm

0 0.0000 0.0001 0.0109 0.0000

1 0.0182 0.1877

2 0.0481 0.3488

3 0.0589 0.3735

4 0.0393 0.2401

5 0.0000 0.0000

0 1.1444 0.002 41,3011

1 1.0802 39.1359

2 0.9033 33.1447

3 0.6458 23.9867

4 0.3373 12.5752

5 0.0000 0.0000

0 0.0221 0.00004 47.6269

1 0.0237 45.1908

2 0.0160 38.3341

3 0.0090 27.7811

4 0.0058 14.5782

5 0.0000 0.0000

0 1.8909 0.003 42.1592

1 1.7912 40.0250

2 1.5213 33.9750

3 1.1033 24.6372

4 0.5777 12.9335

5 0.0000 0.0000

0.0006 0.1433

0.0823

Table 7 (Continued)

- - El X10-^ E, X10-^

0 3.6502 0.006 29.8813 0.052

1 3.4646 28.3779

2 2.9433 24.0990

3 2.1357 17.4816

4 1.1208 9.1794

5 0.0000 0.0000

0 4.3610 0.008 14,4312 0.025

1 4.1445 13.7596

2 3.5234 11.6473

3 2.5584 8.4523

4 1.3441 4.4395

5 0.0000 0.0000

0 1.5600 0.003 5.8537 0.010

1 1.4825 5.5611

2 1.2600 4.7248

0.9146

0.4805

0 . 0 0 0 0

5,4231

0 . 0 0 0 0

Table 8. The calculated results for absolute errors and Euclidean norms for equal spacing in a axially reflected core at different radial levels

Nine-point

cm cm X 10 -3 Over all

Five-point

X 10 Over all norm

0 0.0000 0.005 0.0075 0.0000

1 0.0963 3.3350

2 0.3689 1.3922

3 0.8128 3.0825

4 1.2746 5.1571

5 4.6798 7.2482

6 1.1765 2,3044

7 0.0000 0.0000

0 1.4785 0.004 51.7290

1 1.5423 50.1119

2 1.6462 45.3807

3 1.7773 37.8684

4 2.0271 28.0321

5 1.7173 16.4291

6 1.3216 6.3143

7 0.0000 0.0000

0 0.7875 0.003 55.9785

1 0.8650 54.1634

2 1.0006 48.8033

3 1.1736 40.2747

4 1.2854 29.1294

5 1.2626 16.0606

6 0.8785 6.2885

7 0.0000 0.0000

0.01 0.0172

0 = 106

Table 8 (Continued)

Nine-point

X 10 -3 Over all

Five-point

E. X 10 1

-3 Over all norm

0 0.4123 0.001 43.4105

1 0.3193 41.9898

2 0.1200 37.7737

3 0.1514 31.0564

4 0.4468 22.2801

5 0.7045 12.0068

6 0.5447 4.7354

7 0.0000 0.0000

0 1.0144 0.002 22.4297

1 0.9377 21.6938

2 0.7507 19.5043

3 0.4761 16.0128

4 0.1431 11.4507

5 0.2436 6.1130

6 0.2317 2.4182

7 0.0000 0.0000

0 . 0 8 2

Table 9. The calculated results for absolute errors and Euclidean norms for equal spacing in axially reflected core at different axial levels

Nine-point

X 10 -3 Over all

Five-point

X 10 -3 Over all

0 0.0000 0.002 0.0075 0.0000

1 1.4785 51.7290

2 0.7875 55.9785

3 0.4123 43.4105

4 1.0144 22.4297

5 0.0000 0.0000

0 0.0963 0.002 3.3350

1 1.5423 50.1119

2 0.8650 54.1634

3 0.3193 41.9898

4 0.9377 21.6938

5 0.0000 0.0000

0 0.3689 0.002 1.3922

1 1.6462 45.3807

2 1.0005 48.8033

3 0.1200 37.7737

4 0.7507 19.5043

5 0.0000 0.0000

0 0.8128 0.002 3.0825

1 1.7773 37.8684

2 1.1736 40.2747

3 0.1514 31.0564

4 0.4761 16.0128

5 0.0000 0.0000

0.091 0.0172

0 . 0 8 8

0 , 0 6 6

Table 9 (Continued)

Nine-point

X 10 -3 Over all

Five-point

X 10 -3 Over all

0 1.2746 0.003 5.1571

1 2.0271 28.0321

2 1.2854 29.1294

3 0.4468 22.2801

4 0.1431 11.4507

5 0.0000 0.0000

0 4.6798 0.005 7.2482

1 1.7173 16.4291

2 1.2626 16.0606

3 0.7045 12.0068

4 0.2436 6.1130

5 0.0000 0.0000

0 1.1765 0.002 2.3044

1 1.3216 6.3145

2 0.8785 6.2885

3 0.5447 4.7354

4 0.2317 2.4182

5 0.0000 0.0000

0 . 0 2 8

-3 »ie

5 O 2.

A Five-Point

# Nine-Point

0 —.id a S 4 Isâial Sistfflaee (cm)

Fig. 10. Absolute error as a function of radial distance for z = 0 and equal spacing in a cylindrical bare reactor core

40 • o m

A Five-Point

@ Nine-Point

\ \ \ \.

I 2 ) 4

Radial distance (cm)

xlO 56

48 h k O u <0

o CO ja <s

A Five-Point

® Nine-Point

\ 2 3 4 Radial distance (era)

12. Absolute error as a function of radial distance for z = 2 and equal spacing in a cylindrical bare

reactor core

#40 r4 O a § 32

A Five-Point

# Nine-Point

2 3 , .4 Radial distance (cm)

Fig. 13. Absolute error as a function of radial distance for 2 = 3 a n d e q u a l s p a c i n g i n a c y l i n d r i c a l b a r e reactor core

- 3 xlO

k O u «48

3 r-f 0 1 •*°

0 1 2 3 4 5 Radial âlstanee (cm)

^ Five-Point

# Nine-Point

Z:4 --e— f ^•

•3 '

r-=0 'A \

Five-Point

Nine-Point

0 «SS

^ Asial dis&nes (em) *

Fig. 15. Absolute error as a function of axial distance for r = 0 and equal spacing in a cylindrical bare

reactor core

- 3 «10

40 k 4) Q)

Five-Point

Nine-Point

t -=a--r:1

• — — - —- —- - A- —

2 3 Axial distance (cm)

Fig. 16. Absolute error as a function of axial distance for r = 1 and equal spacing in a cylindrical bare

reactor core

-3 xlO

A Five-Point

® Nine-Point

2 3 4 Axial diotimce (cm)

Fig. 17. Absolute error as a function as axial distance for r = 2 and equal spacing in a cylindrical bare reactor core

k k 40

"3 32 n

A Five-Point

•"=3

r- •» s I

2 3 4 Axial dlstaace (cm)

Fig. 18.

reactor core

- 3 xlO

24 •

a Pive-Polnt

* Nine-Point

— r : 4

2 3 4 jysisl iistKïc® (es)

Fiq. 19. Absolute error as a function of axial distance îor r = 4 and equal spacing in a cylindrical bare

reactor core

S « S

« 20 3

Pive-Polnt

® Nine-Point -

0 êi. 1 3 4 9 6

Radial âietasco (es)

Fig. 20. Absolute error as a function of radial distance for 2= 0 and equal spacing in a radially reflected cylindrical reactor core

I II 1 I I r

i( , !

A Five-Point

0 Nine-Point

\ % < \ * > \ \ * > A ' \

i % \ \\ \

3 3 4 5 6

Radial âiotanee (os)

Fig. 21. Absolute error as a function of radial distance for z = 1,2,3, and equal spacing in a radially reflected cylindrical reactor core

- 3 X10

12 U o k U

® 10 3

Five-Point

Nine-Point

r-—«•-.

-p-- - _z#c. JL

3 4 9 6

Fig. 22. Absolute error as a function of radial distance for z = 4 and equal spacing in a radially reflected

cylindrical reactor core

-3 xlO

(4 2 6 h m

3 3 •3 5 m

1 1 1 r 1 r

g Pive-Point

a Nine-Point

0 M". - "f i L

3 4 5 6 Axial distance (cm)

Fiq 23. Absolute error as a function of axial distance for r = 0 and equal spacing in a radially reflected

cylindrical reactor core

'4 Pive-Point

Nine-Point

•» 3

3e \ r»t

r.| (». ^

•—6-.

3 4 « 6 Axial dletonce (em)

Fig, 24. Absolute error as a function of axial distance for r = 1 and equal spacing in a radially reflected cylindrical reactor core

& Five-Point

@ Hlae-Point

L •3

H,2 \ \

\ \ \4

» \ , \ 16 -

•I -

«5 »

• r»4

i-iS. . r.î «U - -

% \ > \ •\ 'A

3 4 9 6 Axial dletmnce (em)

Fig. 25. Absolute error as a function of axial distance for r = 1 , 3, 4, 5 and equal spacing radially reflected cylindrical reactor core

-3 xlO

t » 3 o «M 9 M O S

Pive-foint

Nine-Point

Axial distance (cm)

Fig. 26. Absolute error as a function of axial distance for r = 6 and equal spacing radially reflected cylindrical reactor core

-I 1 1 " 1 1 " "1

\ r. 4

\ ^ Five-Point

@ Nine-Point

\ / \ e - \ / \

/\ /r»0 \/

/ / r.4

fr.O \ \ \

w » #- » 1

3 4 8 6 7 Asial dlstenee (oq)

27^ Absolute error as a function of axial distance for r = 0, 4 and equal spacing in an axially reflected cylindrical reactor core

-3 x10

56 A Five-Point

h # Nine-Point

z \ \ r4 O (0 AO

L 32 h

a* w \

3 4 5 6

Axial distance (cm)

Fig. 28. Absolute error, as a function of axial distance for r = 1, 2 , and equal spacing axially reflected cylindrical reactor core

u « 24

A Pive-Point

ô Nlne-Polnt

\ 88 I- \

I \ \ \

\ i- - - 4 â -? -9 ^ 0 1 2 3 4 5 6 7

Axial dletence (en)

Fig. 29. Absolute error as a function of axial distance for r = 3 and equal spacing axially reflected cylindrical reactor core

2:6 t \

Pive-Point

Nine-Point

/ / / ;

\ \ w 'À

Jk JL i T g— '

\ 1 3 4 5 6

30. Absolute error as a function of radial for 2= 0 and equal spacing axially reflected cylindrical reactor core

3 -3 xlO xlO

Ô Ïive-Point

64 I- @ Nine-Point

u o L #4 «1 « 3 rH 0 1 40

\ L.--^

Badial distance (cm)

. 31. Absolute error as a function of radial distance for z = 1, 5 and equal spacing assially reflected cylindrical reactor core

-I 1 1 1 1 r

A A4 -

\ yio I \

I ^ Plve-Polnt

f.:. \ \ @ Nine-Point

il U 'k'\ \\

- h / ^ \ \

I I " r , : / \

« t- li! \ V\

4 u \% \\\ 1

Radial dieianoe (cm)

Pig. 32. Absolute error as a function of radial distance for z = 2, 3f 4 and equal spacing axially reflected cylindrical reactor core

Five-Point

Nine-Point

3 "glO

20 29 SO 39 40 45 50 55 60 0 B Iteration

Fig. 33. Residual vector norm as a function of iteration and equal spacing in a cylindrical bare reactor core

Five-Point

Nine-Point

0 % 10 15 ÏQ a» 30 36 40 45 50 36 60

Iteration

Fig. 34. Residual vector norm as a function of iteration and equal spacing in a radially reflected cylindrical reactor core

ï 1ô'

I £ 10»

® Nine-Point

10 19 30 39 30 39

Iteration

40 49 90 99 @0

Fig. 35. Residual vector norm as a function of iteration and equal spacing in an axially reflected reactor core

Similar calculations were performed considering nonequal

(Figs. 7-9) spacing along radial and axial axes for bare

cylindrical cores, radially reflected cylindrical cores, and

axially reflected cylindrical cores. The neutron flux values

as calculated by the nine-point, the five-point, and the

analytical techniques are shown in Tables 10-12. The absolute

errors and Euclidean norms for the radial and axial cases are

calculated and listed in Tables 13-18. Plots of absolute

error versus radial and axial distances are shown in Pigs. 36-

53. Also, plots of the residual vector norm versus iteration

are shown in Figs. 54-56.

From the plots of Figs. 36-53 one can observe the

accuracy of the nine-point formulation over the five-point

formulation in comparison with the analytical solution. The

over all Euclidean norms of the nine-point and five-point

formulas are 0.0239 and 0.0724 (bare cylindrical core), 0.032

and 0.057 (radially reflected cylindrical core), and 0.032 and

0.065 (axially reflected cylindrical core), respectively.

These results indicate the accuracy of the nine-point formula

tion over the five-point formulation for these cases.

The convergence rate as calculated from the slopes of the

curves of Figs. 54-56 for the nine-point and five-point

formulas are 0.4672 and 0.3026, 0.3002 and 0.2049, and 0.349

and 0.248; respectively. These results indicate that the nine

point formula converges faster than the five-point formula.

Table 10. The of

flux distribution in the = 1.0 cm, = 0.33 cm"

core for unequal spacing •1, and R = z = 5 cm

and for input data

Method r z

0 0.5 1.2 2.1 3.4 5

ANS 0 1.000 0.986 0.919 0.761 0.435 0.000

0.5 0.988 0.974 0.907 0.751 0.430 0.000

1.2 0.930 0.916 0.854 0.707 0.405 0.000

2.1 0.790 0. 779 0.726 0.601 0.344 0.000

3.4 0.482 0. 475 0.443 0.367 0. 210 0.000

5 0.000 0.. 000 0.000 0. 000 0.000 0.000

NFS 0 1.000 0.987 0.923 0.769 0.445 0.000

0.5 0.984 0.971 0.909 0.758 0.439 0.000

1.2 0.924 0.912 0.854 0.712 0.412 0.000

2.1 0.785 0.774 0.724 0.604 0.350 0. 000

3.4 0.478 0.4 71 0.441 0.367 0.213 0.000

5 0 .000 0.000 0.000 0.000 0.000 0.000

FPS 0 1.000 0.980 0.920 0.774 0.453 0.000

0.5 0.990 0.970 0.910 0.766 0.449 0.000

1.2 0.9 39 0.920 0. 864 0.727 0.426 0. 000

2.1 0. 808 0.792 0.744 0. 626 0.366 0.000

3.4 0.501 0.491 0.461 0.388 0.227 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000

Table 11. The flux distribution in a radially reflected core for unequal spacing and for input data of D =0.5 cm, E = 0.1385 cin~l, R = z = 5 cm, D_ =1.0

_1 c c Kç, cm, = 0.1 cm , and b = 2 cm

Method r z 0

1.2 to

3.4 5 5.9 7

ANS 0 1.000 0 .989 0.9 37 0. 813 0. 548 0.160 0.075 0.000

0.5 0.988 0.977 0.925 0.803 0.541 0.159 0.074 0.000

1.2 0.930 0.920 0.871 0. 756 0.509 0.149 0. 069 0. 000

2.1 0.790 0.781 0.740 0.642 0.433 0.127 0.059 0.000

3.4 0.482 0. 476 0.451 0.392 0.264 0.077 0.036 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NPS 0 1.000 0.988 0.938 0.817 0.556 0.168 0.078 0.000

0.5 0.984 0.972 0.922 0. 803 0.546 0.165 0,077 0. 000

1.2 0.922 0.911 0.864 0.752 0.512 0.155 0.072 0.000

2.1 0.781 0. 771 0.731 0,636 0.433 0.131 0.061 0.000

3.4 0.475 0,468 0.444 0.386 0. 263 0.080 0.037 0.000

5 0.000 0-000 0.000 0.000 0.000 0.000 0.000 0.000

FPS 0 1.000 0.9 79 0.925 0. 803 0.541 0.151 0.070 0.000

0.5 0.9 84 0.964 0.911 0.791 0.532 0.149 0.069 0.000

1.2 0.924 0.904 0. 855 0.742 0.499 0.139 0.065 0.000

2.1 0.785 0. 768 0.726 0.630 0.424 0.118 0.055 0.000

3.4 0.480 0.469 0.443 0.385 0.259 0.072 0.034 0.000

5 3.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Table 12. The flux distribution in the axially reflected core for unequal spacing and input data of = 0.5 cm, = 0.1536 cm~^, R = z = 5 cm, = 1.0 cm,

= 0.1, and b = 2 cm

Method z r 0 0.5 1.2 2.1 3.4 5 5,9 7

ANS 0 1.000 0. 991 0.946 0.837 0.593 0.192 0.091 0.000

0.5 0.986 0.976 0.932 0.825 0.584 0.189 0. 090 0.000

1.2 0. 918 0.910 0.867 0. 769 0.544 0.175 0.084 0. 000

2.1 0., 761 0. 754 0. 720 0.637 0.451 0.146 0.070 0.000

3.4 0.435 0.431 0.412 0.364 0.258 0.084 0. 040 0. 000

5 0., 000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NPS 0 1.000 0.987 0.939 0. 828 0.585 0.184 0.089 0.000

0.5 0 .985 0.972 0.924 0.815 0.574 0.184 0.087 0.000

1.2 0. 918 0.907 0. 862 0.760 0.535 0.172 0.081 0 . 000

2.1 0.763 0.753 0.716 0.631 0,444 0.143 0.068 0.000

3.4 0.440 0.435 0.413 0.364 0.256 0.083 0.039 0. 000

5 0.000 0.000 0.000 0.000 0.000 0.000 0. 000 0.000

FPS 0 1.000 0.988 0.940 0.829 0.583 0.181 0. 086 0.000

0.5 0.973 0.961 0.915 0.807 0.567 0.176 0.084 0.000

1.2 0.905 0.894 0.851 0.750 0.527 0.164 0.078 0. 000

2.1 0.754 0.744 0.708 0.625 0.439 0.136 0.065 0.000

3.4 0.437 0.431 0.410 0.362 0.254 0.079 0.038 0.000

5 0.000 0. 000 0.000 0.000 0.000 0.000 0.000 0.000

Table 13. The calculated results of absolute errors and Euclidean norms for unequal radial spacing

Nine-point

X 10 a. Over all norm

Five-point

E^ X 10 -3 Over all

0 0.0000 0.014 0.0239 0.0000

0.5 0.9925 5.4548

1.2 4.4811 1.5076

2.1 8.4843 13.3880

3.4 10.1004 17.0610

5 0.0000 0.0000

0 3.4117 0.0116 1.4338

0.5 2.0372 3.4939

1.2 1.7118 3.2665

2.1 6.2204 14.7191

3.4 8.7662 18.7148

5 0.0000 0.0000

0 5.3882 0.0108 8.9705

0.5 4.1133 3.7182

1.2 0.4191 9.6519

2.1 4.2172 19.3912

3.4 7.3282 20.8571

5 0.0000 0.0000

0 5.0212 0.0093 18.1284

0.5 4.5887 13.4563

1.2 1.5651 17.8663

2.1 2.5360 24.6107

3 = 4 5.5770 / .

22.4860

5 0.0000 0.0000

0 3.4133 0.0061 19.5374

0.5 3.4992 16.5201

1.2 1.7094 18.9820

2.1 0.9048 21.5734

3.4 2.9908 17.5557

5 0.0000 0.0000

0.0152 0.0729

0.0244

0.0316

0.0405

0.0423

Table 14. The calculated results of absolute errors and Euclidean norms for unequal axial spacing

Nine-point Five-point J.

X lO"^ . Over all 2 norm

X 10"3 . Over all 2 norm

0 0 0.0000 0.014 0.0239 0.0000 0,015 0.0729

0.5 3.4117 1.9705

1.2 5.3882 8.9705

i-H CM

5.0212 18.1284

3.4 3.4133 19.5374

5 0.0000 0.0000

0.5 0 0.9925 0.007 5.4548 0.023

0.5 2.0372 3.4939

1.2 4.1133 3.7182

2.1 4.5887 13.4563

3.4 3.4992 16.5201

5 0.0000 0.0000

1.2 0 4.4811 0.005 1.5076 0.028

0.5 1.7118 3.2665

1.2 0.4191 9.6519

2.1 1.5651 17.8663

3.4 1.7094 18=982

5 0.0000 0.0000

2.1 0 8.4843 0.012 13.3880 0.043

0.5 6.2204 14.7191

1.2 4.2172 19.3912

2.1 2.5360 24.6107

3.4 0.9048 21.5734

5 0.0000 0.0000

3.4 0 10.1004 0.017 17.0610 0.044

0.5 8.7662 18.7148

1.2 7.3282 20.8571

2.1 5.770 22.4860

3.4 2.9900 17.5557

5 0.0000 0.0000

Table 15. The calculated results for the absolute errors and Euclidean norms for unequal spacing in a radially reflected core at different axial levels

i. Over all norm

0 0.0000 0.012

0.5 5.5330

1.2 8.3670

2.1 3.7412

3,4 8.0554

0.032 0.0000 0.022

9.8640

11.6643

9.5740

7.0512

5 7.6262 9.4980

5.9 3.4505 4.5772

7 0.0000 0.0000

0 4.1068 0.017 3.5372

0.5 4.6682 13.2392

1.2 3.1232 14.8633

2.1 0.2216 12.3833

3.4 5.5649 8.9519

5 6.8353 9.9392

5.9 3.0811 4.7800

7 0.0000 0.0000

0 8.0393 0.016 5.9306

0.5 8.6475 15.0606

1.2 7.1029 16.4963

2.1 3.512 13.8742

Table 15 (Continued)

Nine-point

X 10 -3

Five-point

i . Over all norm

3.4 2.6604

5 5.7403

5.9 2.5744

7 0.0000

2.1 0 9.5645 0.019

0.5 10.6341

1.2 9.4261

2.1 6.0955

3.4 0.0501

5 4.4338

5.9 1.9461

7 0.0000

3.4 0 6,9750 0.0145

0.5 8.1708

1.2 7.6274

2.1 5.5059

3.4 1.2635

5 2.4241

5.9 1.0739

7 0.0000

9.9463

9.7861

4.6997

0 . 0 0 0 0

5.5013 0.027

13.2954

14.5336

12.2742

8.8018

8.4194

4.0422

0 . 0 0 0 0

2.4525 0.0150

7.2482

8.0739

6.8192

4.9316

5.0145

2.4101

0 . 0 0 0 0

Table 16. The calculated results for absolute errors and Euclidean norms for unequal spacing in a radially reflected core at different radial levels

Nine-point

X 10 -3

Five-point

a. Over all norm

0 0 0.0000 0.015 0.032 0.0000 0.012

0.5 4.1068 3.5372

1.2 8.0393 5.9306

2.1 9.5645 5.5013

3.4 6.9 750 2.4525

5 0.0000 0.0000

0.5 0 5.5330 0.014 9.8639 0.027

0.5 4.66882 13.2392

1.2 8.6475 15.0606

2.1 10.6341 13.2954

3.4 8.1708 7.2482

5 0.0000 0.GOOD

1.2 0 0.8367 0.014 11.6693 0.030

in o 3.1232 14.8633

1.2 7.1029 16.4963

2.1 9.4261 14.5336

3.4 7.6274 8.0739

5 0.0000 0.0000

3.4 0 8.0554 0.010 7.0512 0.018

0.5 5.5649 8.9519

1.2 2.6604 9.9463

Nine-point

E. X 10-3 «2

Five-point Over all norm

2.1 0.0501 8.8018

3.4 1.2635 4.9316

5 0.0000 0.0000

0 7.6262 0.013 9.498 0.020

0.5 6.8353 9.9392

1.2 5.7403 9.7861

2.1 4.4338 8.4194

3.4 2.4241 5.0145

5 0.0000 0.0000

0 3.4505 0.006 4.5772 0.009

0.5 3.0811 4.7800

1.2 2.5744 4.6997

2.1 1.9461 4.0422

3.4 1.0739 2.4101

5 0.0000 0.0000

Table 17. The calculated results for the absolute errors and Euclidean norms for unequal spacing in an axially reflected core at different radial levels

a. Over all norm

0 0 0.0000 0.017

0.5 3.3114

1.2 6.8140

2.1 9.1620

3.4 7.7172

5 8.0772

5.9 2.4637

7 0.0000

0.5 0 1.0691 0.018

0.5 4.4104

1.2 7.9860

4Ù» ± J.U .

3.4 10.0221

5 5.8732

5.9 2.9078

7 0.0000

1.2 0 0.0562 0.016

0.5 3.2333

1.2 6.7322

2.1 9.3664

0.032 0.0000 0.019

3.0821

6.0966

8.1434

9.7077

11.3705

5.3258

0 . 0 0 0 0

12.3577 0.039

15.2753

17.7134

18.4130

16.9178

13.5052

6.3452

0 . 0 0 0 0

13.1323 0.040

15.8654

18.1159

18.8638

Nine-point

X 10 -3

Five-point

H. Over all norm

3 . 4 9 . 4 4 4 7 16.8525

5 4.6853 12.9399

5.9 2.5704 6.0842

7 0.0000 0.0000

0 2.2877 0.011 6.8954

0.5 0.4168 9.2383

1.2 3.5298 11.3365

2.1 6.1349 12.2482

3.4 6.8002 11.7533

5 3.3133 10.0476

5.9 1.9236 4.7225

7 0.0000 0.0000

0 5^054.2 0.0067 1.3631

in o 3 . 4 2 8 5 0.0595

1.2 1.4008 1.5368

2.1 0.6080 2.6653

3.4 1.9292 3.6903

5 1.1605 4.8132

5.9 0.8743 2.2579

7 0.0000 0.0000

Table 18. The calculated results for the absolute errors and Euclidean norms for unequal spacing in an axially reflected core at different axial levels

Nine-point

E. X 10-3 E. X 10-3

Five-point Over all norm

0 0 0.0000 0.006 0.032 0.0000 0.019

in o 1.0691 12.3577

1.2 0.0562 13.1323

2.1 2.2877 6.8954

3 . 4 5.0542 1.3631

5 0.0000 0.0000

0.5 0 3.3114 0.007 3.0821 0.025

0.5 4.4104 15.2753

1.2 3.2333 15.8654

2.1 0.4168 9.2383

3.4 3.4285 0.0595

5 0,0000 0.0000

1.2 0 6.8140 0.013 6.0966 0.029

0.5 7.9860 17.7134

1.2 6.7322 18.1159

2.1 3.5298 11.3365

3.4 1.4008 1.5368

5 0.0000 0.0000

2.1 0 9.1620 0.018 8 . 1 4 3 4 0.030

in o 10.4298 18.4130

1 . 2 9,3664 18.8638

Nine-point

X 10 Over all norm

Five-point ^ T Over all

2.1 6.1349

3.4 0.6080

5 0.0000

3.4 0 7,7172 0.017

0.5 10.0221

1.2 9.4447

2.1 6.8002

3.4 1.9292

5 0.0000

5 0 8.0772 0.012

0.5 5.8732

1.2 4.6853

2.1 3.3133

3.4 1.1605

5 0.0000

5.9 0 2.4637 0.005

0.5 2.9078

1.2 2.5704

2.1 1.9236

3.4 0.8743

5 0.0000

12.2482

2.6653

0 . 0 0 0 0

9.7077 0.029

16.9178

16.8525

11.7533

3.6903

0 . 0 0 0 0

11.3705 0.025

13.5052

12.9349

10.0476

4.8132

0 . 0 0 0 0

5.3258 0.012

6.3452

6.0842

4.7225

2.2579

0 . 0 0 0 0

-3 xlO

Five-Point

# Nine-Point

B - ; r "O I /

\ ; \ \ \ »

I \ 11 I \ é I

i V : I / \ /

I \ I \ '• \ \ \ 1

i 3 4 5 6

Radial disteace (cm)

Fig. 36. Absolute error as a function radial distance for z = 0 and unequal spacing in a cylindrical bare reactor core

xlO 2 0

6 Five-Point ^

@ Nina-Point' \ / \

\ * «0.5 /

I • 10 V 9 •3 I

I I à 1

/ Z'Oa \ \

\ ) \ \ \

2 3 4 9 Radial distance(cm)

Fig. 37. Absolute error as a function of radial distance for z = 0.5 and unequal spacing cylindrical bare reactor core

.3 X*)

10 A Pive-Point

@ Nine-Point

\ I r > '

y''\ /

Zil.S \ \

3 - / \ ' w

3 4 9 6 7 Baâial dtstaaoe (œ)

Pive-Point

* ® Nine-Point

\ . \ > 'J

8 @ 4 5 . e Radial dlatoBoe (cm)

*10 as

/ \ so - *«3.4,

r i'°

# A Plve-Point 1 ' 9 Nlne-Polnt

I I s 4 6 6 7

Radial âiatsaee (em)

Fig. 40. Absolute error as a function of radial distance for 2 = 3.4 and unequal spacing cylindrical bare reactor core

*10 20

I I 10

1 1 r— r' -• •

• ; ; • ' 1 1 1

Plve-Polnt

Nine-Point .

• ; ; • ' 1 1 1

' 1 ' -

f \ \ 1

/ ' / ' -

\ • -'A \

/' K.. 1 1 —J 1— _B —1 —

3 4 5 6

Axial dlBtanes (era)

Fig. 41. Absolute error as a function of axial distance for r = 0 and unequal spacing cylindrical bare reactor core

& Piv©>=Point

a Nine-Point

K' ' \ /

/-M,' A" M

'na.i * 1 >

À ' \ \

'ril a

; v.;i ' I •' 1

I •••O.S

\ ! \r . i .A \ I <: •

% V / ' \ /

'1 V^r.M ^

_i s 4 9 e 7 Aslal â&stsaoo (ea)

Fig. 42. Absolute error as a function of axial distance for r = 0.5, 1.2, 2.1, 3.4 and unequal spacing cylindrical bare reactor core

$ 3 M 0

^ Pive=Polnt

@ Nine-Point

z=0/ \ / \

"s. / /(

• I t I I

Z : 3.4

\ Z =3.4

\ \ \ \

\\\ . \ \\

v:% iL'

4 5 6 &i5 uBZlCG \Cw/ Radial diataTPS

43. Absolute error as a function of radial distance for z = 0/ 3.4 and unequal spacing in a radially reflected cylindrical reactor core

*10 16

U Five-Point

Nine-Point

0) ® 10 3 r4 o tn 5 .

\ Z = 0 5 \

z =0.5

4 « •

\ / \ I

I \ \ \

\\ \ \ N\ \\

3 4 5 6 Radial distance (cm)

Fig. 44. Absolute error as a z =0.5 and unequal cylindrical reactor

function radial distance for spacing radially reflected core

I S10 :

^ Five-Point

# Nine-Point

\ \ II II fK / \ II

Z:1.2 -A

e ( r i f l !

\ . 2.1 \

V\ \ \

/••

/is 2.1

% 3 4 5

Pig. 45. Absolute error as a function of radial distance for 2 =1.2, 2.1 and unequal spacing radially reflected cylindrical reactor core

PïO.5

> m Five-Point

# Nine-Point

'T "\\

..^<0 II 7

: i r7 /

a/ /r.o

3 4 5 6 Azial distance (cm)

46. Absolute error as a function of axial distance for r = 0, 0.5 and unequal spacing radially reflected cylindrical reactor core

.3 xlO

1 0 -r,1.2 / \ f

Five-Point

Nine-Point

k k 12 G

I s 10 5

\ \ I ^""=2.1

/ / "A \

' '\ \\ 6

/ J r:2.1

}( /'\ y \ / N

1 r:3.4 ^.

Axial distance (cm)

Fig. 47. Absolute error as a function of axial distance for r = 1.2, 2.1, 3.4 and unequal spacing radially reflected cylindrical reactor core

XI0 16 T-

I 11! H 0

1 10 "

r.B.9 r:9

Five-Point

e Nine-Point

k % rsS-S" XX \

3 4 5 G

kzisJL distance (cm)

Fig, 48. Absolute error as a function of axial distance for r = 5, 5.9 and unequal spacing radially reflected cylindrical reactor core

jk A Five-Point -3 H

xio ^ % Nine-Point

X \ \ \ \

/ 1® [- /

3 L \ \

/ t \\ 1/

I 0 1 g 3 4 5 s

Âsiol distame® (om)

Fig. 49. Absolute error as a function of axial distance for r = 0, 0.5 and unequal spacing in an axially reflected cylindrical reactor core

-3 xlO

T— r

A Pive-Point

# Nine-Point

o 0) :

12 rs2.1

/ rs1-2 ^ —

rr2.1 \

\ •v'/

\ \ \ /

r s3*^

/ \/ ,ar'

î ,/ A

2 3 4 5 6 Axial SiStance (cm)

50. Absolute error as a function of axial distance for r = 1.2, 2.1, 3.4 and unequal spacing axially reflected cylindrical reactor core

3 K10 20

V'' \ \ * -m »

\ \ « 12 ^ 17^=0

W • \

A Pive-Polnt

a Nine-Point

\ / A \x ^^.0 A/

3 4 5 6 Radial distance (cm)

Fig. 51. Absolute error as a function of radial distance for z = 0, 0.5 and unequal spacing axially reflected cylindrical reactor core

.S tiiO

î , . $ ?

Pi.1.8

A i.i.a / \

\\ \ W

Pive-Point

Nine-Point

\ \ \ \\

3 4 5 6

Fig. 52. Absolute error as a function of radial distance for r = 1.2, 2.1 and unequal spacing axially reflected cylindrical reactor core

-S «10

:»3 4

Five-Point

# Nine-Point

• I \ I \ Xt9

- y \ \ \

•\ \ .::3 4

\ »iS.9 k . \ \

j'9-9 \ \

3 3 4 5 6 Radial diotaace (om)

Fig, 53. Absolute error as a function of radial distance for r = 3,4, 5/ 5.9 and unequal spacing axially reflected cylindrical reactor core

Five-Point 1

Nine-Point

• 2 to

• 9 0

10 ao 29 30 S9 0 40 43 90 99 eo

Iteration

Fig. 54. Residual vector norm as a function of iteration and unequal spacings cylindrical bare reactor core

s g iS* *» u

"3 I lô'

@ Pive-Point

@ Nine-Point

_L _L _L. _L. J_ _L _L

10 19 30 29 30 30 <0 49 90 99 60

Iteration

Fig. 55. Residual vector norm as a function of iteration and unequal spacing radially reflected cylindrical reactor core

o e f 10®

Pive-Polnt

Nlne-Folnt

10 ie 30 25 30 S9 4o 45 50 95 GO

Al*At4 AR

Fig. 56. Residual vector norm as a function of iteration and unequal spacing axially reflected cylindrical reactor core

A. Completely-Reflected Cylindrical Core

In this case, the completely-reflected cylindrical core

is considered as shown in Fig. 57. Equal and unequal spacing

along the radial and axial axes are considered as shown in

Figs. 58-59, and 66-67. The results for this case may be

arranged as follows:

1. Equal spacing

The neutron flux values for this case are calculated

using the nine-point formula and the five-point formula and

tabulated in Tables 19-20. Graphs for the neutron flux versus

radial (z = 0) and axial (r = 0) distances are shown in Figs.

60-63. Also, graphs for the residual vector norm versus

iteration are shown in Figs. 64-65.

The purpose of this calculation is to show the validity

of the nine-point formula for a completely reflected

cylindrical core. The analytical solution for this kind of

calculation can not be determined. No comparison has been

made between the nine-point formula and the five-point formula

with respect to accuracy.

The convergence rates as calculated from the slope of the

curve of Figs. 64-65 for the nine-point formula are 0.459 and

0.397 and for the five-point formula are 0.299 and 0.287,

respectively. These results indicate that the nine-point

formula converges faster than the five-point formula.

Fig. 57. Completely reflected cylindrical reactor core

Fig. 58. Equal spacing with more spacing along the radial axis for a completely reflected cylindrical reactor core

Equal spacing with more spacings along the axial axis for a completely reflected cylindrical reactor core

Table 19. The neutron flirx distribution in a completely reflected cylindrical core for equal spacing along the radial and axial axes and input data of =

0.5 cm, Z = 0.1385 cm D = 1.0 cm, Z = 0.1 cm ^

Method r z 0 1 2 3 4 5 6 7

HPS 0 1.000 0.954 0.829 0.638 0,407 0.162 0.067 0. 000

1 0.939 0. 896 0.778 0.599 0,382 0.153 0.063 0.000

2 0.762 0.727 0.632 0.487 0.311 0.125 0.052 0.000

3 0.491 0.468 0.407 0.315 0.203 0.083 0.035 0.000

4 0.155 0.152 0.133 0.103 0.067 0.036 0.018 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

PPS 0 1.000 0.915 0.782 0.595 0.373 0.144 0. 059 0. 000

1 0.938 0.858 0.733 0.558 0.350 0.135 0.056 0.000

2 0.759 0.695 0. 594 0.452 0.284 0.111 0.046 0.000

3 0.486 0.445 0.380 0.290 0.184 0.074 0.031 0.000

4 0.152 0.1387 0.119 0.091 0.060 0.032 0.015 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Table 20. The neutron flux distribution in a completely reflected cylindrical core for equal spacing along the axial and radial axes and input data of =

0.5 cm, = 0.1536 cm = 1.0 cm, = 0.1 cm ^

Method ^ 0 1 2 3 4 5 6 7

0 1.000 0.962 0.851 0.676 0.448 0.181 0.077 0.000

1 0.926 0.891 0.788 0.626 0.415 0.172 0.071 0.000

2 0.726 0.699 0.619 0.491 0.327 0.136 0.057 0.000

3 0.441 0.425 0.376 0.299 0.201 0.086 0.037 0. 000

4 0.129 0.124 0.110 0.088 0.060 0.034 0.017 0.000

5 0.000 0.000 0.000 0. 000 0.000 0.000 0.000 0.000

0 1.000 0.962 0.850 0.673 0.444 0.179 0.076 0.000

1 0. 863 0.830 0.733 0.581 0. 383 0.155 0.066 0.000

2 0.658 0.633 0.560 0.444 0.294 0.120 0.051 0.000

3 0.391 0.376 0.333 0.264 0,176 0.074 0.032 0.000

4 0.108 0.104 0.092 0.073 0.051 0.028 0.014 0.000

5 0.000 0.000 0.000 0.000 0.000 0. 000 0.000 0.000

rivs-Folnt

@ Nins-Point

5 4 9 6 7 Sediel dist^ee (ea)

Pig. 60. Neutron flux as a function of radial distance for z = 0 and equal spacing completely reflected cylindrical reactor core with more spacings along thé radial axis

11 T r =1

9 . \ \

@ yiTa>Polnt

@ Nine-Point

I . Î

^ f »o

9 4 5 iaiol d&stoaoo (ea)

Fig. 61o Neutron flux as a function of axial distance for r = 0 and equal spacing in a completely reflected cylindrical reactor core

10^ \ 's.

V s s c

® Pivo-Point

® Hino-Polnt

W \ \ \ \ \

\ \ . s s 0

\ w ;\

W \ % \

4 S Radial dlatenoo (on)

Fig. 62. Neutron flux as a function of radial distance for z = 0 and equal spacing in a completely reflected cylindrical reactor core

-1 HfO

0 Pivo-Polnt

® Hlae-Point

I. \ \ \ \ \ \

M S I s

a 4 s e Axial diatenoe (cm)

Fig. 63. Neutron flux as a function of axial distance for r = 0 and equal spacing in a completely reflected cylindrical reactor core with more spacings along the axial axis

Il • ir I I I I t I I

# Five-Point

.0 10 J I u J I I L

0 5 % 15 ao 25 30 5S 40 45 30 %

Iteration

Fig. 64. Residual vector norm as a function of iteration and equal spacing in a completely reflected cylindrical reactor core with more spacings along the radial axis

I u 10

^ ,ô'

Pive-Polnt

e Nino-Point

_L _L _1_ J L

10 19 30 89 30 as 40 49 SO 39 80

Iteration

Fig. 65. Residual vector norm as a function of iteration and equal spacing in a completely reflected cylindrical reactor core with more spacings along the axial axis

2. Unequal spacing

The neutron flux values for this case were calculated

using the nine-point formula and the five-point formula and

tabulated in Tables 21-22. Graphs for the neutron flux versus

radial (z = 0) and axial (r = 0) distances are shown in Figs.

68-71. Plots of residual vector norm versus iteration are

shown in Figs. 98-99.

This calculation was performed in order to show the

validity of the nine-point formula for the completely reflected

cylindrical core calculation where the analytical solution can

not be determined. No comparison was made between the nine-

point formula and five-point formula with respect to accuracy.

The convergence rates as calculated from the slopes of

the curves of Figs. 72-73. For the nine-point formula the

rates are 0.279 and 0.333 and for the five-point formula the

rates are 0=191 and 0.234. Therefore, these results indicate

that the nine-point formula converges faster than the five-

point formula.

I 1 30

Fig; 66. Unequal spacing mesh with more spacing along the radial axis for a completely reflected cylindrical reaetor core

z <f> = C )

Unequal spacing mesh with more spacing along the axial axis for a completely reflected cylindrical reactor core

Table 21. The neutron flux distribution in a completely reflected cylindrical core for unequal spacing along the radial and axial axes and input data of D =

-1 -1 0.5 cm,, = 0.1385 citi , = 1.0 cm, = 0.1 cm

Method g ^ 0 0.5 1.2 2.1 3.4 5 5.9 7

0 1.000 0.988 0.938 0.816 0.555 0.164 0.075 0.000

0. 5 0.975 0.963 0.914 0.796 0.541 0.160 0.073 0.000

1. 2 0.882 0.871 0.826 0.720 0.490 0.146 0.067 0.000

2. 1 0.674 0.664 0.629 0.548 0.375 0.115 0.053 0. 000

3. 4 0.235 0.235 0.224 0.196 0.135 0.055 0.028 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0 1.000 0.979 0.924 0.802 0.539 0.147 0.067 0.000

0. 5 0.975 0.954 0.901 0.782 0.525 0.143 0.066 0. 000

1. 2 0.880 0.861 0.813 0.706 0.475 0.131 0.060 0.000

2. 1 0.666 0.652 0.616 0.535 0.361 0.103 0.048 0.000

3. 4 0.218 0.214 0.202 0.177 0.122 0.048 0.024 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0. 000 0.000

Table 22. The neutron flux distribution in completely reflected cylindrical core for unequal spacing along the axial and radial axes and input data of D =

-1 -1 0.5 cm, = 0.1536 cm , = 1.0 cm = 0.1 cm

Method ^ 0 0.5 1.2 2.1 3.4 5 5.9 7

0 1.000 0.987 0.939 0.828 0.584 0.172 0.081 0.000

0.5 0.976 0.963 0.916 0.808 0,568 0.172 0.078 0.000

1. 2 0.874 0.862 0.820 0.723 0.508 0.156 0.071 0.000

2.1 0.642 0.634 0.603 0.532 0. 375 0.119 0.055 0.000

3.4 0.201 0.198 0.189 0.167 0.119 0.051 0.027 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0 1.000 0.986 0.936 0.823 0.574 0.169 0.077 0.000

0.5 0.956 0. 943 0.895 0.787 0.550 0.162 0.074 0.000

CM H 0.846 0.835 0.793 0.698 0.488 0.145 0.067 0.000

2.1 0.614 0.605 0.576 0.507 0.356 0.109 0.051 0.000

3.4 0.169 0.166 0.158 0.140 0.101 0.043 0.023 0.000

5 Cl.000 0. 000 0.000 0.000 0.000 0.000 0.000 0.000

® Pive-Point

® Nlne-Foint

\\ \\"0

-j u a s <3 s s

Radial dietanoe (om)

Fig. 68. Neutron flux as a function of radial distance for z = 0 and unequal spacing in a completely reflected cylindrical reactor core with more spacings along the radial axis

10 ® Plva-poiaî

® Nln«-Polnt

I -9 •i

3 4 i 6 Axial distance (cs)

Fig. 69. Neutron flux as a function of axial distance for r = 0 and unequal spacing in a completely reflected cylindrical reactor core

® PlYe-Polnt

® Nine-Point

I \\ *»0

3 4 Radial distance (en)

Fig. 70. Neutron flux as a function of radial distance for z = 0 and unequal spacing in a completely reflected cylindrical reactor core

-1 kIO

\ \ r . \ r.o

@ Plve-Polnt

® Nine-Point

3 4 s s Axial distance (cm)

Fig. 71. Neutron flux as a function of axial distance for r = 0 and unequal spacing in a completely reflected cylindrical reactor core with more spacings along the axial axis

I I ,6'

@ Pive-Polnt

@ Nine-Point

_L I I I I >

0 10 20 30 40 90 SO 70 80 90 100 110 120

IteeatiOB

Fig, 720 Residual vector norm as a function of iteration and unequal spacing in a completely reflected cylindrical reactor core with more spacings along the radial axis

k .2 ! 10

# Five-Point

@ Nine-Point

_i J L 1.. J L

0 10 Î0 30 40 50 00 70 60 90 100 ^^0

Fig. 73. Residual vector norm as a function of iteration and unequal spacing in a completely reflected cylindrical reactor core with more spacings along the axial axis

IV. CONCLUSIONS

The technique of formulating the one-group time independ

ent neutron diffusion equation in cylindrical geometry in

terms of the nine-point finite difference equations resulted

in more accuracy than the five-point formulation. The accuracy

of the nine-point formulation over the five-point formulation

was illustrated for a variety of configurations in reactor

physics problems. The configurations include a bare reactor

core, radially reflected reactor core, and axially reflected

reactor core. The accuracy of the nine-point formula over the

five-point formula is not affected by the spacing along the

radial and axial axes. This accuracy was measured by the

absolute error and Euclidean norm criteria.

The nine-point formula converges faster than the five-

point, formula for all calculations concerned with this reactor

physics problem. This was illustrated by computing the slope

of the curve from the plots of residual vector norm vs. itera

V. SUGGESTION FOR FURTHER STUDY

The nine-point finite difference formulation technique

for the one-group static neutron diffusion equation in two

spatial dimensions (r-z) cylindrical geometry exhibited

increased accuracy and convergence rate over the five-point

finite difference formulation for regions with homogeneous

Dirichlet boundary conditions for physical surface ('f'suj.-Face ~

other boundaries. The nine-point finite difference formula

tion may be readily extended to the neutron diffusion equation

in three dimensions (x, y, z) in rectangular geometry.

The present work may be extended to related fields deal

ing with the Helmholtz equation in r-z cylindrical geometry

(for example the nonhomogeneous equation (V«D (r, z) Vcj) (r,z) -

E(r,z)(j)(r,z) + S = 0). Further work may be accomplished with

the mixed boundary conditions which stated in physical terms

means no inbound neutron current (or possibly an albedo

boundary).

0) and homogeneous Neumann condition on the

VI. BIBLIOGRAPHY

1. H. Greenspan, C. N. Keller, and D. Okrent, eds., "Computing Methods in Reactor Physics," Gordon and Beach, Science Publisher, Inc., New York, 1968.

2. Eugene L. Wachspress, "Iterative Solution of Elliptic Systems," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966.

3. D. J, Hughes, J. E. Sanders, and J. Horowitz, eds., "Physics and Mathematics," Pergamon Press, New York, 1958, Vol. 2.

4. A. Rohach, "Application of a Nine-Point Relation to the Eigenvalue Problem," Trans. Am. Nucl. Soc., 17, 232 (1973).

5. J. A. Nohel and W. P. Timlake, Presented in Proceedings of Second International Conference on the Peaceful Uses of Atomic Energy, Vol. 16, United Nations, Geneva, 1958.

6. W. G. Bickley, "Finite Difference Formulae for Square Lattice," Quart. Mech. Appl. Math., 1, 35-42 (1948).

7. D. Greenspan, "On "Best" Nine-Point Laplace Difference Analogues on Rectangular Grid," Quart. Mech. Appl. Math., 12, 111-116 (1959).

8. Ward C. Sangren, "Digital Computers and Nuclear Reactor Calculations," John Wiley and Sons, Inc., New York, 1960.

9. J. M. Clark and K. F. Hansen, "Numerical Method of Reactor Analysis," Academic Press, New York, 1964.

10. I. S. Sokolnikoff and R. M. Redheffer, "Mathematics of Physics and Modern Engineering," McGraw-Hill, Inc., New York, 1966.

11. Robert V. Meghreblian and David K. Holmes, "Reactor Analysis," McGraw-Hill Book Company, Inc., New York, 1960.

12. S. D. Conte and Carl de Boor, "Elementary Numerical Analysis," McGraw-Hill, Inc., New York, 1965.

13. D. Young, "Iterative Methods for Solving Partial Difference Equations of Elliptic Type/" Trans. Am. Math. Soc., 76, 92-111 (1954).

14. H. B. Keller, "On Some Iterative Methods for Solving Elliptic Difference Equations," Quart. Aop. Math., 16, 209-226 (1958).

VII. ACKNOWLEDGMENTS

The author wishes to express his deepest appreciation and

gratitude to Dr. Alfred F. Rohach for his mature advice during

this work. Recognition and gratitude should be given to the

government of Iraqi Republic for their financial support

during the study period. Sincere gratitude is given to my

wife, Nidhal, for her encouragement throughout my graduate

study.

VIII. APPENDIX A: TAYLOR SERIES EXPANSIONS

OF THE FLUX ABOUT A POINT (r\,Zj)

The expansions of the flux at the points

(zj-l'Zj)' and (rj'Zj.i) in terms of

(ijfZj) are

*1+1 = ^1

2 3 *1 = + RDr*o + f- D^^o + f- + Oth^) (A-1)

4i,j-l = *3

2 3 *3 = *o - t Dr*o + r Dr*o - r 'A-Z'

*1,]+! - ^2

= è + T D,*n T 2 + h + °lh*l (A-3)

4i,i-l - *4

*4 = *o - B Dz*o + h °^0 §- + O(h^) (A-4)

The expansions of the

(i+l,j+l), (i-l,i+l),

the point (i,j) are

flux at the corner point,

(i-l,j-l)f and (i+l,j-l) about

*1+1,]+! -

3 4 = E — (RD^ + è

m=0 m! ^ ^ °

•^ = to + + TD^to

+ Î + I T^Os*o +

+ I + i + 5

+ & *T'»r:s*o

+ O(h^) (A-5)

By adding and subtracting (j)^ to the right hand side of Eq. 5

and by substituting for the resulted terms from Eq. A-1 and

Eq. A-3, the result is

•i+l,j+l = •i+l.j + •i,j+l - *o *

+ I R'TDfrzt)

+ O(h^) (A-6)

Eq, A-6 may be written as

- *1 - *2 + *o = ™rzfo + I + J rzz* o

+ O(h^) (A-7)

4i-i,i+i =

4^ = Z (-LD + TD )* é m=0 m!

4^ = *0 - LDr*o TDz*o

+ I I^Dr*o + I - LTD^I-o

- i L'Dr*o + i T^D:*o + ? '™rrz*o " I '°rzz*c=

+ O(h^) (A-8)

By adding and subtracting (j)^ to the right hand side of Eq. A-8

and by substituting for the resultant terms from Eq. A-2 and

Eq. A-3, the result is

+ 0(h*) (A-9)

Equation A-9 may be expressed as'

- *3 - *2 + fo = - " I ^rrz'l'o ' I

+ Ofh^) (A-10)

*i-l,i-l = 4»^

= I — (-LDI - BCi (j)^ m=0 m!

= *0 - LD^ o - ®°Z O

+ I + I + «''rzl'o

- I - k °2»0 - I t'sOfZ^O - I ''^\zz*0

+0(h4) (A-11)

By adding and subtracting (j)^ to the right hand side of Eq.

A-11 and by substituting for the resultant terms from Eq. A-2

and Eq. A-4, the result is

= *1-1, j - *0 + '('o " I ^®°rrz^o

- I + O(h^) (A-12)

Equation A-12 may be written as

4^ - *3 " *4 + *0 = ^^^rz'ô " t ~®°rrzô ' I "^"rzzô

+ O(h^) (A-13)

^i+l,j+l ^ 4)

3 ** = Z — (RD^ - BD )* (t)^

m=o m !

4 * = *0 + RDpOo - BDgOg

+ I * J

+ I - i - 1 + I '^^Drzz'fo

+ O(h^) (A-14)

By adding and subtracting 4)^ to Eq. A-14 and by substituting

for the resultant terms from Eq. A-1 and Eq. A-3, the result

- *4 + *o = - I + 1

+ O(h^) (A-15)

IX. APPENDIX B; SIMPLIFICATION OF DERIVATIVE TERMS

1. Simplification of the partial derivatives and

R T at the points (r^ + Z-Zj), (r^ - z-Zj), (r-r^, z^ + »

B and (r-r^,Zj " 2^•

a. From Eq. 12

+ f (z-2y)Dfr2*o * I

2 Let a = Offo + I Dr*o + r °r*o

2 3 or A = i + §_ + f- dJ+o'

R"^ 3 By adding and subtracting (j)^ and -g— to the right side,

the result is

2 3 A = i {•„ + + f- X>li^ + f-

- *0 + #- Dr*o " f" Dr*o)

From Eg. A-1

*l = *0 + + #- °^o f-

r=ri+|

A = -V^ - §4 Dr^o (B-1)

Let F = Dp + § Drrz'f'o z

From Eq. A-15

- *1 - *4 + Oo = - I è RB^Drzzto

+ Oth^)

. . . . I

From Eq. A-7

pi - *1 - *2 + *o = *-°rz*o + &

+ I RT^D^ (f)^ + O(h^) zz

Thus, •[_ _ * - - *2 + *o Tn A (B-2') ^ RT 2 rzz^o

C = (B-3) Let <- = I^xzz'Po

Therefore, Eq. 12 may be written as

9 è 3r

r=ri+|

= A + (z-Zj)F + (Z-Zj) C (B-4)

b. From Eq. 13

r=ri-2

= Dr*o • 2 Mo + (s-SjiDrzto + 8~ Vo

- I (Z-Z.)D^ 0,22*0 rz

A' = 0,40 - I Oft) + r Oft)

or A' = - è + f - D;*o - 5- 0:*-)

_2 ^ 3 .

rô 2 rô rô

3 By adding and subtracting (j)^ and to the right side, the

result is

2 = ' è '*0 • + 2" °r*o

8 03 r-o - (| )_ ) y j

From Eq. A-2

*3 = *o - LDrto I" °r^o ' I" ^

A' = - '®-5'

^ ^rz^o "" 2 ^rrzto

From Eq. A-10

<l> - ^3 - ^2 + Og = LTDpgOg + 2 L TDj-j-gCl)^ 2 ^rzztg

+ 0(h)

F = 4> " 2 - #2 ^

LT 2 ^rzz^o (B-6)

From Eq. A-13

W - YT - YA + 4), i L^BD •„ - i lb2D..,^„ . *- w 6

+ O(h^)

(B-6')

2^ 3r L

r=ri-2

= A' + (z-Zj)F' + (Z-Zj) C (B-7)

c. Prom Eq. 14

3 é 3z T

Z=Zj+j

Dz*o + (r-ri)Of2*o + I 02*0

1 >2 i' "rrz^o + 2

2 + I (r-i\)Dr22*o + §- Da^o

Let s = »z*o • I °^o + h 2 3

G = i (TD,*o + §- + |-

3 By adding and subtracting (j)^ and to the right side,

the result is

G = & (*o + + I" Oz*o ^ i" °z*o + i~ Dz*o " I"

From Eq. A-3

„2, . t' „3^ , „,^4, +2 = *o + TOs+o + t + r °z*0 + 0<h )

G = *2 T O - §4 (B-8)

Let H = + 2 0,22*0

From Eq. A-7

- *1 - *2 + to = ^Drz*o i B^TDrrz*o ^ I '^^^rzz'f'o 1 r.n,2r

+ O(h^)

Thus, -

H = - *2 + 1 RT 2 ^ °rrzto (B-9)

From Eq. A-10

4> " 3 " 2 to " - LTOfs*. + I ^•"'rrz'^o - I I'T^Drsz*o

+ O(h^)

H = (j) - (j) - (j)2 + (f>

LT '° + i t °rrzfo (B-9')

Let ^ 2 ^rrgto (B-10)

H 3z ^T

z=Zj+2

G + (r-r^) H + (r-r^) K (B-11)

d. From Eq. 15

a* 7z = + (r-r.)D^2(j)^ - I Dg*o + j

B Z - Z . - j

- I (r-zï)Drzz4o + f" °z'^o

" ''z^o - 2 "zTo ' 8^ "z^o

2 G' = - I D^è + S- D^é

G' = - I (-B + §- - |- D^*.)

gS 3 By adding and subtracting and to the right

side, the result is

2 3 G' = - 5 l*o " BDatg + |- - f - - tg

+ r o!*. - r D^+nl

From Eq. A-4

^4 ~ ""zô ' 2~ zô 6 "zô

2 j 4u = + ~ D^è - #- D^é + O(h^)

=' = - - h »:*o '=

' = Dr to - 2 °rzz*o

H • = h *o - &

From Eq. A-13

4^ - *3 " *4 + *0 = - I - j LB^D^^^cf)^

+ 0(h)

H' = 4^ - *3 - *4 + *o 1 LB + 2 L Drrz*o (B-13)

From Eq. A-15

^ - *1 - #4 + = - - Î

+ I RB'^D^ + O(h^) 4,

THUS ,

(B-13')

9(1) G' + (r-r^)H' + (r-r^) K

B 2=Zj-5

(B-14)

2. Simplification of P, Q , and M.

P = r? - (r^ +

P = r? - rj - riR - |-

.2 P = - r^R - (B-15)

Q = r: - (r, + f)3

3 Q = ~ f~ (B-16)

M = rf - (i\ +

4 - 4 - -i # - #4 - ''='i f J.D

M = -2rf R - 3 2„2 2 "•

(B-17)

c: Evaluation of Q - j r^P

By substituting for P and Q from Eg. B-15 and Eg. B-16,

respectively, the result is

T 8 - I V = - Î

d; Evaluation of j M - •! r^Q + i r?P

By substituting for P, Q, and M from the Equations (B-15,

B-16, and B-17), respectively, the result is

1 „ . I r.Q + i rjp = - r.R= - ij (B-19)

e: Evaluation of [R(^ Q - ~ M - j r^ Q + r?P)]

By substituting for Q - from Eq. B-18 and for

(j M - J r^Q + Y Zi?) from Eq. B-19, the result is

R(| Q - i r.P) = i r.R^ - ij (B-20)

R(| Q - I r.P) - (l M - I r.Q + i r^P) = - L r.R^ -

(B-21)

3. Evaluation

I DilR(i Q - I r.P) - (| M - I r.Q + I r2p)]D^^,4.^

By substituting for [R(^ Q - ^ ^ ~ J

1 4 from Eq, B-21 and lumping the term - D^R with the

truncation error O(h^), one has

i D^[R(| Q - 1 r.P)-(i M - § r.Q + i r2p)l= - |j D^r-R^^^^^

(B-22)

By substituting for P from Eq. B-15 and then lumping the

term - with the truncation error O(h') one has JL Z O

Thus, the terms that include the derivatives in Eq. 17 may be

written as

ID = + i D^[R(§ Q - I r.P)-(| M - I r.Q + I

By substituting for the first and second term in B-24 from

Eq. B-22 and Eq. B-23, one has

ID = - ^

- k - h h^i ' \ z z^o

ID = - 1% D^r. RT (TD^4,^ + RDJ^^)

4. Simplification of P', Q', and M'

P' = Ir, - |)2 - rl

2 T ^ 2 P' = r. - r.L + _ r.

,2 P' = -i\L + (B-26)

Q' = (r, - |)3 - rl

Q' = - BrJ I + 3ri

Q' = - ^ r? L + (B-27)

M' = (r^ - ~

M' = - Arl | + - 4r. |- +

4 M' = - 2r? L + I r? 1? - ^ r. ^ (B-28)

1 / 1 Z 1 10

d: Evaluation of [(i M' - 4 r,Q' + % r?P')+L(^ Q' - r^P')] O X ^ 1 . « / 6 , ^

By substituting for P', Q', and M' from the equations

(B-26, B-27, and B-28) one has

1 M' - I r.Q' + I r.P'S- - r.

(B-29)

- |r.P') =|r.

[(i M- - I r.Q' + i r.PM + L(| Q' - I r.PMI

5. Evaluation

- & Dz':? M' - § fiO' |r.P')+ L ( |q' -

By substituting for [(j M' - j r^Q' + j r^P')+L(y Q' - r^P')j

1 4 from Eq. B-30 and lumping the resulted term DgL D^rz^o

with the truncation error O(h^) one has

- I "al'l"' - I fiO' + I V + 0' - I riP'l:°rrz*o

By substituting for P' from Eq. B-26 and lumping th

resulted term Do L^T^D^cj) with the truncation error O(h^), 192 2 z O

m = - Î8 ( B - 3 2 )

Thus, the terms that include the derivatives in Eq. 19.may be

written as

IE = - §4 Drrct)

+ k °2V' "rzz^o

IB = - is Dzfi LTlTD^o "

54 Vi'I'' °rrz*o - ''rzz^ol '®-")

By substituting for P' from Eq. B-26 and lumping the

1 2 2 3 / 4 V resulted term - D^B L with the truncation error 0 (h ),

one has

- 38 = & 03ri*'L»3*=

I ">3"?™' - + i ZiP'l + LI? 0' - & riP')l°rrz*o

By substituting for t(j M' - r^Q' + •^ r? P ' ) + L ( ^ 0' - r^P')]

from Eg. B-31, one has

I D j K I M ' - § r.Q- + i r?P')+L(i Q' - | r.P• ) 1

= 54 Djr.L^ (B-35)

Thus, the terms that include the derivatives in Eg. 21 may be

written as

IG . L

+ 14 »3ri»^»r:z*o

:G = ?â Djr^ LBIBC^O + LDr*o)

+ ijDjr.CL^ Drrz*o +

- Î8 P Dz*o

By substituting for P from Eg. B-15 and lumping the

12 2 3 resulted term - D^B R D^cj)^ with the truncation error

O(h^), one has

- Î8 •>4 = ià »3 fi (B-371

I D^t(i M - I r.Q + i r2p) - E(| Q - i r.P)lD„^4,„

By substituting for tM - ^ r^Q + r?P) - Rfy Q - ]

from Eq, B-22, one has

i D^n| M - I r.Q + I r^P) - R(| Q - I

= 54 °4 °rrz*o l»-:»'

Thus, the terms those include the derivatives in Eq. 23 may be

written as

IK = is D4 ^ °:*o

- ij ri B

+ Is »4 fi \ / o

- iy D4 r. D,::**

IH = I3 D4 ri R B (B - R

^ 54 °4 Wo - Wo' (B-39)

X. APPENDIX C: APPLICATION OP

2 %A THE IDENTITY V (j) = (}) n

1. Simplification of the following terms;

- is Dl fi + B °r*o)

b: - is 0% fi LTI' ^

c: + r. LB(B + L 0^4,^)

d: + D4 ri RB(B D^*o - ®

The above terms have appeared in relations 25, 33, 36, and 39

of Appendix B, respectively. These terms may be further

simplified by using the following techniques :

- CT^cf) = 0 (C-1)

where = •— , n - 1,2,3,4 n

2 The Laplacian operator 9 may be expressed in cylindrical

geometry (r-z coordinates) as

= (D^ + + L D^)

- ij Di r. ET(Td34,o RD^to)

This could be simplified by using the following identity:

(RDf + = (RD^ + TDg)(DJ + 0% +

2 By substituting for \7 (|)^ from Eq. C-2, one has

(KDf + TDs)°l*o = + ™rzz + Î7 »r*o

+ TD^„4.„ + TDz*o + Î7 »r:*o

T 03*0 + R rr^o = (R D, + T

- Ir (RD^^o + TDr^t*)

- lKDr,:$o + TDrrst,) (C-31

By multiplying both sides of Eq. C-3 by - r^ RT, one has

- ?8 °1 + ™r*o' = -T8 °1

+ h + ig Djr.RKED^^^t^ +

The term - %g r^ RT(RD^ + that appeared in Eq.

C-4 may be simplified using the following technique;

From Eq. À-1 and Eq. A-3

.2. . r.3. ^ •l = *0 + ®Vo + 5- »r*o + r °r*o +

2 3 *2 = *0 + + f- + h Oz*o + °'h ) By adding the equations then multiplying both sides of

resulted equation by - r^ RT and then lumping the

terms of orders O(h^) and higher with the truncation error

O(h^), one has

- 35 »i ?! RKBVo + = - Is »i fi *1

+ 0(h*) (C-5)

By substituting for - r^ cr., RT (RD^ q + TD^^g) in Eq.

C-4 from Eq. C-5, one has

- Di r. RKTD^ + = - 3? °1 fi °1 + •2 - 2*0'

+ & °1 RT(K>^o +

+ Di r. RT(ED^,,t„ + TDrrz*o)

+ O(h^) (C-6)

Using the same technique as in (a) the terms in (b), (c),

and (d) may be simplified. Thus,

- 38 °2 = - h °2V2 '^'•2 + *3 - 2*0'

+ k °2 LT'Ta, Z

+ O(h^) (C-7)

+ Î5 Ojr. + LD^4,„) = - I5 + *3 - 2*^)

- k °3 + LDX)

- h Ojf i tB 'BDrfa+o + ^°rzz»o>

+ 0(h*) (C-8)

ij D r. RB(BD 4, - . - ij Djr.o, RE(4. + - 24, )

- h °4 RBtBD, - RD^*^) z

- i? D, r. RBCBDrr:** -

+ 0(h*) (C-9)

By adding Eq. C-6, C-7, C-8, and C-9, one has

R, - Î8 - -"A' R.

+ is D3r.LB(BD^0„ + LD'I,^) R-

+ Î8 04=1*8(503*0 - ED^O^)

' • 38 Vl *'"*1 + *2 - •o'

• # °2°2 LT(*2 + *3 - ^•o'

- & 0,0, LBl*, + 4.3 - 24^1

- h D4O4 •'i •^'•4 + *1 - 2*0' + Î8 °1 + ™rz*o>

+ & °2 •o - L»r*o) z R,

- & D3 L»(»»r *0 + LD;*g) Z

Ï8 D4 =®<®°r *0 - R»r*o' z

+ ts Di:%RT(RDr2,*o + ""rrs^o' R,

+ W DzfiLTlTDrrzOo " '''rzz^o' R.

- Dgr.LBlBDrrsOo + LOrz o)- 04=^1'®'^^rrz^o ' R,

(C-10)

The terms including the derivatives in Eq. C-10 may be

rearranged as follows;

R. R, R.

- 38°4®®\Z'''O

isfiriTR'OrzzOo Î8°2VI''Vz+o IçDjr.BL^^^Oo

+ j8l>4riBR-Dr,A R,

R. + iiPzriLT^Drrzt,

R, IjDjriLB^Drr.+o

(C-11)

The evaluation of the derivatives, ' rz^o

D <f) rrz^o R

for n - 1,2,3,4, may be performed by

\ using the flux expansions as cited in Appendix A and then

applying the continuity of neutron current density between

adjacent regions. Also, the terms that are of orders O(h^)

and higher have to be lumped with the truncation error 0(h^).

Hence, the following formulations have been obtained:

R, D^LR(RH-L) + ®2'^3 " *0" +

(C-12)

Rg 2 ELLR(R+L) [LDlt + l " <^0^ ' *o)] +

(C-13)

»r*o DgLRfR+L) - y + RD3($3 " 1 + 0(h)

(C-14)

"1*0 D.LR<R+L) 'to,'*! - *.1 + 3% " •o"

(C-15)

^rz^o 2 :

R, 2X.RTD., [DiL(*--*i-*2+*o)-D2R(*"-*3-*2+*o)] "(h)

(C-16)

Drz+o = 2L&D: [DiL(*l-*i-*2+*o)-D2R(*2-*3-*2+*o)] + Rç 2

(C-17)

^rz^o [D3R(4^-*3-44+*ol-D4L(*4-*i-44+*o)l + 0(h)

(C-18) R3 3

"rz+o R>

= miBD^ [D3R(*^^*3-*4+*o)-D4L(**-*l-*4+*o)] 0(h)

(C-19)

Drrz^o D^{R+L)T + 0(h)

(C-20)

^rrz^o DjlR+Lyr D, 1 ^2 2 ^(4) -(i)^-(i)2+(Po)+ L-(<î> + 0(h)

(C-21)

^rrz^o R-d^TL+RTB

D D, ^(())^-())3-(j)^+(j>^)+ pr^4^-4i-*4+0o) + 0(h)

(C-22)

^rrzto ^4

(L+R) B o 4

— ((j) -4^-#4 + *o)+ ^(4) + + 0(h)

(C-23)

°rzz^o D^RfT+B)

D D 4,.4 + 0(h)

(C-24)

^rzz^]

D ^((i)^-(|)3-())2+(j)Q)+

3,.3 + 0(h)

(C-25)

^rzz - 2

R3' B^Ht+BT

B4RTTÎB)

^9 2 3 ;^((|) -<|)3-(j)2+^Q)+ g-((|) ~<f'3~'J'4+<f'o^

^1 1 ^4 4 — ((j) -4^-02+^0)+ g-(')) ~<l>]_°<f'4+(|>Q)

+ 0(h)

(C-26)

+ 0(h)

(C-27)

2. Evaluation

IK = - il Dlfi <®\rz*o + °rzz»o'

h Bi'Drrzto " T^Or:z»o'

+ Î4 °4=^i (K^»rrz*o ' ®'°rzz*o' R>

(C-28)

The terms in Eq. C-28 may be arranged as

IK = - 34 fi*'

1 Jz - D R.

'2 W n O - n

9(j) __c

3 dz I R.

9z D, - D

2 f 36,

9z 3 9r - D

9<!)(.

2 ÏÏF

R, (G-29)

By applying the continuity condition of the neutron current

Ho density, i.e., = D

4 3z o , D R, 2 Tz

= D R,

3 W R,

D 3<()(

1 8F = D

3<|)ç

2 ÏÏF , and D 9(J)^

= D R.

4 9r , one has

IK = 0 (C-30)

By substituting for R ' ^rz^o R

' ^rrz^o R n n n

in Eq. C-11 from the Eqs. C-12, C-13, (C-25), and (C-27),

respectively, and adding the result to Eq. C-30, one has

IN + IK = - r^RT + *2 " 2*o)

- Î8 °2°2 ""i +*3 -

- iê °3°3 '•4 + *3 - 2*0)

fa D4O4 RB - 2*0)

+ Ixrs- (R-i) [LD, (*, - ((I,)+RD, (4., - *_)] Z4 ijK 1 i V 6 J G

+ && (R-DILD,**! - *0)+ *03(43 - "»o"

Kl'-Ml

D M -G-O » <1^

MM (T>|

M CO n H-

to U M -e-o

to H H-H3 tr« to

D l-S -e-O

Oil M M M

1-3 ^to a

H •e-O »

Ml M a^\ M 1-1

P-03 to >3 O N

~Sh O » A.

o\| M W M H-tu Nil

O N -e-O » w

to o N -e-O » to

Ml M O^l M M M

to pa o N •e-

a < (U M c 0> ff H-§

toi M •fe-l t^l M Tw a w >0 -e-

w -©-

-e-•P^ + -o-o

I •©" M I -G-

+ -e-o

a M t-i -e-M

-e-to +

+ 0 to w -G-to

w -e-to + -e-

to] M H h3| M CO H-

I tr" a to CD

-e-to I -e-w I -e-

to + -e-

1^ -e-

M I -©-M

to + -e-

-e-1^ I

M •e-

VD M (^1

CO I -G-w -G-

+ •G-o

The derivatives R

ana 0^4.^ can be evaluated using the

flux expansion relations as in Appendix A and lumping all the

terms of orders O(h^) and higher with the truncation error

O(h^). The results may be written as follows

a: TR R,

= TR((j)^ - <j)^) + 0(h*) (C-33)

b; T * Dz*o = TR(({>2 - <|)q) + 0(h*) (C-34)

c: TL' R.

= - TL(*3 - 4)^) + 0(h*) (C-35)

= TLfOg - (f>o) + 0(h*) (C-36)

e: L-B = - LB R,

{(j)3 - (|)q) + 0(h ) (C-37)

f ; -B L D d) z o R-

= BL((})^ - (J)q) + 0(h ) (C-38)

g: BR D^(f)^ = BR(#2 - (jJ^) + O(h^) (C-39)

h: B R = - BR($^ - (Î)q) + 0(h*) (C-40)

By substituting the Eqs. C-33, C-34, , C-39, and C-40,

respectively, into Eq. C-32, one has

= - l6 %l ^i + *2 " 2*0)

- Î6 ^2 i TL(*3 + 4)2 - 2+0)

- %3 ^i BL(*3 + *4 - 2*0)

- Ï6 Z, BR(*i + *4 - 2*g) {C-41)

the formulation and analysis of the nine-point finite

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